Understanding Vectors and Tensors

[Music] hello everybody welcome to the first lecture and in this lecture we are going to discuss about the mathematical concepts that will be required in the coming lectures okay so the main topic starts from lecture two and this lecture we are going to work with vectors and tensors a lot and I will tell you what tensors are okay so this is working with vectors and tensors all let's you start with vectors so we all have learnt about vectors in our schools we see that a vector is something which has magnitude and Direction right so it could be something like an arrow with a with a pointer so the length of the arrow is the magnitude and the direction is the arrow itself isn't it alright that's a basic concept but more important a vector is independent of the coordinate system so you look at a vector from one coordinate system all from other coordinate system or from some other coordinate system the vector does not change so it's independent of the coordinate system think for example of a coordinate system whose basis are e 1 e 2 e 3 the Cartesian coordinate system okay they are perpendicular to each other so it's a Cartesian coordinate system and suppose this vector that we have here we say it is V alright the vector V and it is in the e1 e2 plane making an angle of 45 degree from the even axis all right so this angle is 45 degree so what are the various components of this vector in this coordinate system if I say v1 the component along even then this is the vector V dot with E one right similarly you can get the second component and third component and as this vector is lying in the even e 2 plane so the three components are 1 by root 2 1 by root 2 and 0 isn't it assuming that the vector is of unit magnitude so the components of this vector with respect to this coordinate system is 1 by root 2 1 by root 2 0 you take another coordinate system then the component of the vector with respect to other coordinate system will be something else okay so it's the component of the vector which changes from one coordinate system to the other but the vector itself does not change ok please remember that we have this vector [Music] and I put this big bracket to denote that I am writing the components of this vector as the three entries in a column so this becomes 1 by root 2 1 by root 2 and 0 ok and this is in E 1 e 2 e 3 coordinate system this column that you see here it's the representation of the vector in e 1 e 2 e 3 coordinate system if you choose another coordinate system let us say a red coordinate system where one of the axis is along the vector itself ok so we have e 1 hat e 2 hat and III hat is along III itself so in the new coordinate system the what will be the components the component along even had since human hat is along the vector itself it's one the other two components will be zero than right so in the new coordinate system the representation of that same vector in the new coordinate system even had e 2 hat III hat so that's going to be 1 0 0 right ok so I hope you got some idea that the vector is independent of coordinate system but it's representation or its column form changes from one coordinate system to the other coordinate system alright so I'll just write there ok so you know you could also write it as the vector V is equal to summation over I equal to 1 to 3 VI e I okay the three components summed here or you could write it as summation over I equal to 1 to 3 VI had AI hat so you see when you sum it you get the same vector but individually the components are different alright and you know in this course we will denote vectors by putting a tilde below it now usually we denote vectors by putting a arrow above the vector right like you see here we have put an arrow above the vector but in this course we'll put a tilde below it to say that it's a vector ok so V we will write it as viii with a tilde below and I will tell you why we use this tilde okay all right now what can you do with vectors so there are several operations that you can do with vectors operations with vectors the first one is the usual dot product which all of us know so we have two vectors a and B you see that with the tilde right a dot B so what is it you simply take the individual components multiply them and some it right so this is then summation over AI P I and I equal to 1 2 3 correct you could also write it in some matrix form you know you could write this as a row of a multiplied with the column of B so you have a1 a2 a3 b1 b2 b3 right and this first one is 1 Cross 3 and this is 3 cross 1 when you multiply it you get one cross one right a scalar quantity so dot product of vectors gives you a scalar quantity but there's another thing that you can see now the first one because it's a row which is the transpose of a column right so we could also write it as the vector a transpose multiplied with B so a dot B we could also write as the vector a transpose multiplied with B okay then you have cross product a cross B and I guess all of you know what it gives you so this is basically a 2 B 3 minus a 3 B 1 so this is the even component plus a1 b3 minus a3 b1 that's the e2 component plus a1 b2 minus a2 b1 and that's the e2 component ok the 3 components of this cross product ok now we can also write the same thing in a column form okay so we have column of a cross product with column of B and then the right-hand side we have the three components written in this column a to b3 minus a3 b2 a1 b3 minus a3 b1 a1 b2 minus a2 p1 ok the three components written in this column this is something that you have seen but you know if we can also write it as a matrix times the vector B so we have this column of B b1 b2 b3 and a matrix it's very special matrix which has 0 on the diagonal and the off diagonal components are negative of each other so this is basically a skew symmetric matrix the components of the matrix are formed by the components of a okay for example this first row second column one two first row second column is formed by the third component of a so we have III here first row third column is found by second component of a so whatever is missing you take that component first row third column second is missing so it's formed by second component e to second row third column one is missing so it's found by first component of a II 1 and then the tricky thing is this first one a 3 has to be with a minus sign a 2 remains a 2 and a 1 becomes minus a 1 and then you form this 2 symmetric matrix okay so this is a 3 here we have minus a 2 and here we have a 1 alright so there is a nice way to form this matrix and when you work out this matrix times the column you will indeed see what you have on the left hand side ok the cross product and this matrix so people say that the column of e is actually the axial you know just named axial of this is Q symmetric matrix so whenever you have its two symmetric matrix you can form a column from the three entries in that excuse metric matrix or whenever you have a column you can form it skew symmetric matrix in the way I just told you all right so this is a different way to visualize cross product ok and what you have to remember also that just like the vector which is independent of the coordinate system these operations dot product or cross-product are also independent of the coordinate system for example a dot B is summation of a ibi if you use different coordinate system then it would be summation of AI hat bi hat but the product with the summation has to be the same so dot product will not change likewise for the cross product also so they are geometric they have geometric meaning what is dot product you have two vectors then the dot product is take the magnitude of each of the vectors multiplied with the cause of the angle between the two vectors so it comes from geometry you change the coordinate system magnitude of the vectors do not change the angle between the vectors do not change so the dot product will also not change right so dot product and cross product are basically coordinate free operations when you write it you write it in terms of the components of the vector with change but as a whole it has to be the same there is another interesting operations that one can do with vectors you see dot product gave you a scalar cross product gives you vector this another operation which is called tensor product a tensor product B so the tensor product is denoted by this cross symbol with a circle alright so this gives you what's called a second order tensor ok so this is the first time we are learning about tensor so this gives you a second order tensor so it's second order so we put two tildes below ok second order tensor the way to see this easily is we say whenever we have tensor product and you try to write it in the column form then you take the first vector as a column but the second vector as a row incorrectly reverse of what we did with the dot product in dot product the first one was a row right but here the second one is row so since the second one is a row so therefore it becomes the transpose isn't it it's the transpose of the column first one is 3 cross 1 3 rules 1 column the second one is now 1 Cross 3 so when you do the multiplication what you get is actually a matrix isn't it it's a matrix and the individual components are denoted using two subscripts C IJ okay so C IJ for example the eyath row and jth column what is C IJ when you work this out it is simply a I into B G so you see for the vectors the components have single index subscript whereas for C you have two subscripts there were their first called second order tensor okay and so we use two tilde's 1 tilde for the vector two tildes for second order tensor 3 T loves for third order tensor all right you know we we can also see it that the see was a tense a bit of B right now why don't I expand a as summation over I AI e I ok the vector e and this has to be tensor product with the vector B so that summation over j bj ej correct now we could write the same thing as double summation IJ AI BJ e I tensor product EJ now contrast this with the vector representation for a vector we say a is equal to summation over I AI e I so we have I going from 1 2 3 because it has three basis vectors e1 e2 e3 for second-order tensors the basis vectors are these quantities AI tensor product eg not just II 1 e 2 e 3 with a tensor product so how many of the basis vectors are there there are nine of them I equal to 1 to 3 J equal to 1 to 3 so there are nine basis vectors okay so basis tensors so there are nine basis tenses so any tensor you could write it in terms of these nine basis tensors if you change the coordinate system from e1 e2 e3 to even hat e 2 hat III hat then your basis chances are a I hat tensor product EJ hat okay and the component with respect to death basis is CIJ alright so here this AI into bj is CI j and for a general second order tensor so we write that c is equal to summation over IJ CIJ e I tensor product EJ and C IJ is then the component of D tensor C with respect to the basis tensor AI tensor product ej okay let us see what we get it in the matrix form so this double summation take one of the terms from there let us say c 1 2 e 1 tends to product e 2 now if I write it in the column form the first one will be a column where the second one becomes a row now the column form of e 1 in the e 1 e 2 e 3 coordinate system what will that be that's simply 1 0 0 right because component of e 1 along even is 1 along other directions is 0 so even becomes 1 0 0 whereas e 2 becomes 0 1 0 remember these are the representations of e 1 and E 2 in E 1 e 2 e 3 : system itself if you use some other quarter system to represent II 1 you will get something else you won't get 1 0 0 you have to first find the components in the new coordinate system and then write those components in the column ok anyway so in this coordinate system now you work out this multiplication and what you get you get a matrix 3 by 3 where only the first row second column is nonzero and that is c1 2 rest all are 0 all right so similarly you have several other terms in this representation of tensor and now you can see where is CIJ going so second order tensor when you represent it in a coordinate system you get a matrix okay for vectors you get a column but for tensors you get a matrix and the coefficient with respect to the basis vectors they form the components of the matrix okay and CIJ goes in the I ate row and jth column right you can see it here so I just write here all right and the see with this big bracket so this is the matrix form of the tensor C and the matrix form will change from one coordinate system to the other coordinate system but the tensor itself will not change okay so you could write this same tensor C as double summation IJ CIJ e I tensor product EJ or as double summation IJ CIJ hat of AI hat tensor product EJ hat so two different ways to write the same tensor because we have used two different coordinate system so from here the matrix form has C IJ right whereas from here the matrix form has got C IJ hat so we get two different matrices but the tensor itself is the same okay and just to give a recap the vector with a single tilde then we say it is a first order tensor okay whereas scalars without any tilde it is a scalar and then we say it is a zero its order tensor you put two tilde it's second order tensor you put three tilde it becomes a third order tensor okay and you can keep on thinking of higher order tensors also alright the next thing is how does a second order tensor multiplies with a vector okay alright so suppose we say a is equal to C times B okay so C is the second order tensor B the vector I want to multiply them and I want to see what do I get so I'm going to write the full form of C and B and let's see how do they multiply so the first one see has got double summation over IJ CIJ e I tensor product eg the second one has got a single summation say over K of b k èk and now we have to define this multiplication so we will say that the second row tensor and the first row tensor when they multiply then we take the second vector from the tensor so EJ and ek and take their dot product so that's how we define the multiplication you don't take E I and E K you have to take EJ and EJ take the dot product and let's see what we get so two vectors you take dot product it becomes a scalar right so here then we have triple summation right we have three i j k CI j BK so all these scalar quantities you first write them together and then we have the vector quantities so we have e I tensor product EJ into e K and we know we have to take a dot product of EJ and EJ so this becomes see i j BK ii i of ej dot each a because that's the definition of multiplication so what you get when you dot EG and EK e 1 dot e 1 what is it 1 e 1 dot e to the 0 right towards the perpendicular so when J and K are same it is 1 otherwise it is 0 so there is a very nice function in mathematics which is called Kronecker Delta function and we write it as Delta IJ delta subscript IJ and we say it is equal to 1 if i equal to j and is equal to 0 if i not equal to j ok so we can use this kronecker delta here and it becomes triple summation CI j BK e i delta JK and now you have to see as for the definition if j and k are not the same you get 0 so think of a summation over k so you have k equal to 1 2 3 so that summation will give you a nonzero quantity only when k is equal to j right so basically we can actually get rid of the summation over k and wherever we have k we replace it with j okay so this then becomes summation over IJ CIJ be J CK has been replaced with J times e I and that's your vector a right to begin with we said a is equal to C times B correct and you see carefully if you put your bracket here isn't this a I because vector a you could write it as AI e I so this is basically a ie so what we see is that when we multiply a second-order tensor with a vector we get a vector and the components of the vector are given by this formula so e AI is equal to summation over j CI j bj right if you try to write this in a column form then this is basically the i yet row of C matrix multiplied with the column of b c b1 b2 b3 c i 1c i to see i 3 you multiply it you get what you have here so this actually implies if you want to get the whole column of a a1 a2 a3 then it is simply the matrix c times the column of b so what we have found out finally is that if you want to multiply a second row tensor with a vector the resulting is a vector and to get the column of the resulting vector you simply multiply these C matrix with the column of being usual matrix vector multiplication so I just write here you know with this in hand we can actually go back to our cross product definition do you recall when we had a cross PE then we said in the column form a cross B was actually the matrix times the column of B isn't it so now in the tensor form we can say that the resulting vector C which is a cross B could also be written as second order tensor a times B because second or tensor times vector is equivalent to matrix vector multiplication and now the second order tensor a its column is is this excuse metric matrix a ok and we say that this is a skew symmetric tensor okay I hope all of you are with me so we have to always differentiate between tensor and its representation representation is the matrix which changes from one correlation to the other whereas tensor is independent of the coolant system the other thing is for a vector we know that VI is equal to V dot E I isn't it for a tensor if I let's say I want to extract the coefficient of the tensor what should I do so ckl for example what is its formula okay so we are going to work that out so I will prove you that see KL is equal actually C times e l dot with EK can you see how could you get that formula suppose I want c12 okay let's see for C 1 2 so C 1 2 means C times e 2 dot with E 1 so this is right now a tensor representation let's think of writing this in the matrix form okay so C becomes the C matrix e 2 becomes the column of e 2 and E 1 becomes the column of e 1 so this is same as C matrix the whole C matrix c11 c12 c-13 the whole C matrix multiplied by e 2 and e 2 is then 0 1 0 right you multiply these two you get a vector or a column and then I dot it with E 1 which is 1 0 0 so can you see what you will get within that small bracket you have matrix times this vector 0 1 0 when you multiply you simply get the second column of C matrix ok so you get c1 to c2 to c3 to the second column and then dot with 1 0 0 and you get c1 2 right if you want to extract c1 to this C times e 2 dot with E 1 ok that's the way you can extract the component of a tensor relative to its basis tensor all right similarly you can also show when you multiply two tensors you know multiplying two second-order tensors so there are actually various forms of multiplication one is where you multiply two second or tensors and you get the second order tensor itself a different second order tensor okay so suppose we have C equal to a into B so now we have to see how does this multiplication work out okay so we can expand the whole thing I'm in the right hand side and let's work see what we get so E is then summation over IJ a IJ e I tensor product a J times B which is again double summation over let's say KL of BK l EK tensor product e l right so now let's bring all the scalar things together and the vectors on the other side so we have four summations i g ki l AI j BK l right ii I tensor product EJ multiplied with ek tensor product e L so this multiplication has to be done such that the inner vectors that you have you take the dot product of the inner vectors dot product of the inner vectors which are EJ and EJ here and let's see what we get so you simply have Delta JK right so you have AI tensor product e l CJ and ki are gone and only ein e l are present in to Delta JK now how to work this further ahead so we have two summations one over J + 1 over K and then we have Delta JK so you know what to do right Delta JK is equal to 1 only when J is equal to K so there are two ways to proceed either you remove the summation over J and replace J with K everywhere or you remove the summation over K and replace K with J right just like we were doing earlier but now we have two ways to go ahead because we have summation over both J and K so let us suppose we remove summation over J and replace J with K so we have triple summation i qi l ii i k because J gets replaced with k be k l ii i tensor product e l correct and we can write the same thing as summation over I L and then a summation of a key of AI K BK l e I tensor product e L okay and you see this has to be see the tense of C so don't you think the thing inside the bracket is actually C I and isn't it so this thing inside the bracket is nothing but C I L so now we know what happens when two tensors multiply what happens to its matrix form so CIL is simply summation over K of aik into BK l and can you see what this is really this is actually we take the IH row of a ei1 a I to a i3 so that's the I it row of a matrix multiplied with LS column of B isn't it to get CIL you take the is row of a multiply with LS column of B so to get the full matrix C you then simply have to do a matrix into B matrix so this implies that your C matrix is simply a matrix times the B matrix a very simple formula finally so this means when two tensors are multiplying then the multiplication that we worked out is as if you just multiply the matrix form of the two tensors in the usual way and then the product matrix that you get is the matrix form of the resulting tensor okay very simple so this is about tensors now there will many more things as we proceed in the lecture some new things we may have to learn so that I will address you as we work out the lecture ok there is also something called rotation tensor so this is the physical rotation you have a vector you rotate you get a new vector so we are thinking it's physically but mathematically what should you do on that vector to get the new rotated vector so this rotation is actually done through the rotation tensor okay so you can think of a set of three perpendicular vectors e1 e2 e3 and then it gets rotated into even hat e 2 hat and III hat okay for this rotation there is always a unique rotation tensor so you have a set of tried three vectors and another set of three vectors you can always find a rotation so that it Maps you into the new set of three vectors okay from the first set of three vectors you multiply and you get the second set of three vectors there is a unique rotation to do this operation so we will say that E I hat is equal to R times e I for each of the I equal to 1 2 3 all right and this rotation has got a unique property this rotation tensor it turns out its matrix form is orthogonal matrix okay so are we say it's a Aalto gonal tensor and its matrix form is a orthogonal matrix and such matrices have unique property that is you take R multiplied with its transpose or you take R transpose and multiplied with r then you get identity an identity matrix what does this mean so if we look at the matrix form then we have our matrix multiplied with the transpose of the matrix giving you identity right one one zero zero zero zero zero zero suppose I call this matrix C then what is C IJ so C IJ is nothing but Delta IJ right it's equal to one when I is equal to J that means the diagonal is one off diagonal is zero but you also know from the matrix multiplication you have R into R transpose so we just worked out CIJ when you have two matrices multiplying the resulting matrix coefficient C I L for example is aik bkl right so what will be CIJ so it will be a summation over sum K are I K into R transpose k j the inner index is summed you see the inner index inner index is summed but then our transpose kg is same as our JK isn't it what do we do in transpose we just flip the off diagonal components right so this is same as summation over k RI ki r JK which is equal to Delta IJ can you see its meaning so this says you take the iith rho of r + j throw our and you take the dot product right dot of i a throw and jaea throw that's equal to delta IJ so if you take two different rows of this r matrix and U dot it you get 0 if you take a row and dot it with itself you get 1 so this tells us that the rows of this R matrix are perpendicular to each other similarly you can also prove that the columns of the r matrix are also perpendicular to each other so this rotation matrix is very unique than the rows and columns of this matrix so I'll say two rows any two rows and same thing goes for columns or this matrix are we say orthonormal to each other so basically if the two rows are different it is zero if the two rows are same it is 1 so that means ortho normal ortho for perpendicular normal for 1 normalized okay so that's about the rotation tensor let's see an example of how will the rotation matrix look like so we have this e 1 e 2 e 3 coordinate system and suppose I rotate this coordinate system and think of a new coordinate system even had eetu hat and III hat where III hat is same as III but even hat and E 1 are rotated by theta okay so how will these rotors and matrix look like that's the question here what is this R equal to the components of our now do you recall for a tensor what is R IJ you can go back you see ckl is C times e l dot e K so R IJ is then rotation tensor R into e j dot with E AI but by definition of this rotation tensor are into EJ is nothing but EJ hat dot with E I correct so do you see now what is our I J so the J's column of this rotation matrix is the EJ hat vector written in the even e 2 e 3 corner system can you see it so I have this rotation matrix let's say J is equal to 2 so we have R 1 to R 2 to R 3 2 so our I 2 is simply e 2 hat dot e I so R 1 2 for example is e 2 hat dot e 1 the component of e 2 hat along even R 2 2 is e 2 hat dot e to the component of e 2 hat along e 2 likewise for R 3 2 so this column is basically e 2 hat itself but written in II 1 e 2 e 3 coordinate system that's the column e 2 hat written in even e 2 e 3 coordinate system so for this example can you guess what is e 2 hat from the picture that you see can you guess it so this angle is also theta so when you say e to hat dot e to you get cos theta when you say e to hat dot e 1 you get minus sine theta right it goes in the backward direction so what is then R 1 to e 2 hat dot e 1 that's minus sine theta R 2 to that e 2 hat dot e 2 that's cos theta and r 3 to eat uu hat dot e 3 that's 0 right similarly you can think of the first column of our matrix the first column of our matrix is the column form of e 1 hat in the e 1 e 2 e 3 coded system and the third column of our matrix is the column form of e 3 hat in the e 1 e 2 e 3 colon system so when you work whole everything out we'll finally see that the rotation matrix is simply cos theta minus sine theta 0 sine theta cos theta 0 0 0 1 let's see the third column is simply 0 you know 1 because III hat is same as e 3 ok so that's about vectors tensors rotation and in the next lecture which is actually going to be the first as far as the topic is concerned we will start with the traction vector ok thank you [Music] [Music]

[Music] hello everybody welcome to the first lecture and in this lecture we are going to discuss about the mathematical concepts that will be required in the coming lectures okay so the main topic starts from lecture two and this lecture we are going to work with vectors and tensors a lot and I will tell you what tensors are okay so this is working with vectors and tensors all let&#39;s you start with vectors so we all have learnt about vectors in our schools we see that a vector is something which has magnitude and Direction right so it could be something like an arrow with a with a pointer so the length of the arrow is the magnitude and the direction is the arrow itself isn&#39;t it alright that&#39;s a basic concept but more important a vector is independent of the coordinate system so you look at a vector from one coordinate system all from other coordinate system or from some other coordinate system the vector does not change so it&#39;s independent of the coordinate system think for example of a coordinate system whose basis are e 1 e 2 e 3 the Cartesian coordinate system okay they are perpendicular to each other so it&#39;s a Cartesian coordinate system and suppose this vector that we have here we say it is V alright the vector V and it is in the e1 e2 plane making an angle of 45 degree from the even axis all right so this angle is 45 degree so what are the various components of this vector in this coordinate system if I say v1 the component along even then this is the vector V dot with E one right similarly you can get the second component and third component and as this vector is lying in the even e 2 plane so the three components are 1 by root 2 1 by root 2 and 0 isn&#39;t it assuming that the vector is of unit magnitude so the components of this vector with respect to this coordinate system is 1 by root 2 1 by root 2 0 you take another coordinate system then the component of the vector with respect to other coordinate system will be something else okay so it&#39;s the component of the vector which changes from one coordinate system to the other but the vector itself does not change ok please remember that we have this vector [Music] and I put this big bracket to denote that I am writing the components of this vector as the three entries in a column so this becomes 1 by root 2 1 by root 2 and 0 ok and this is in E 1 e 2 e 3 coordinate system this column that you see here it&#39;s the representation of the vector in e 1 e 2 e 3 coordinate system if you choose another coordinate system let us say a red coordinate system where one of the axis is along the vector itself ok so we have e 1 hat e 2 hat and III hat is along III itself so in the new coordinate system the what will be the components the component along even had since human hat is along the vector itself it&#39;s one the other two components will be zero than right so in the new coordinate system the representation of that same vector in the new coordinate system even had e 2 hat III hat so that&#39;s going to be 1 0 0 right ok so I hope you got some idea that the vector is independent of coordinate system but it&#39;s representation or its column form changes from one coordinate system to the other coordinate system alright so I&#39;ll just write there ok so you know you could also write it as the vector V is equal to summation over I equal to 1 to 3 VI e I okay the three components summed here or you could write it as summation over I equal to 1 to 3 VI had AI hat so you see when you sum it you get the same vector but individually the components are different alright and you know in this course we will denote vectors by putting a tilde below it now usually we denote vectors by putting a arrow above the vector right like you see here we have put an arrow above the vector but in this course we&#39;ll put a tilde below it to say that it&#39;s a vector ok so V we will write it as viii with a tilde below and I will tell you why we use this tilde okay all right now what can you do with vectors so there are several operations that you can do with vectors operations with vectors the first one is the usual dot product which all of us know so we have two vectors a and B you see that with the tilde right a dot B so what is it you simply take the individual components multiply them and some it right so this is then summation over AI P I and I equal to 1 2 3 correct you could also write it in some matrix form you know you could write this as a row of a multiplied with the column of B so you have a1 a2 a3 b1 b2 b3 right and this first one is 1 Cross 3 and this is 3 cross 1 when you multiply it you get one cross one right a scalar quantity so dot product of vectors gives you a scalar quantity but there&#39;s another thing that you can see now the first one because it&#39;s a row which is the transpose of a column right so we could also write it as the vector a transpose multiplied with B so a dot B we could also write as the vector a transpose multiplied with B okay then you have cross product a cross B and I guess all of you know what it gives you so this is basically a 2 B 3 minus a 3 B 1 so this is the even component plus a1 b3 minus a3 b1 that&#39;s the e2 component plus a1 b2 minus a2 b1 and that&#39;s the e2 component ok the 3 components of this cross product ok now we can also write the same thing in a column form okay so we have column of a cross product with column of B and then the right-hand side we have the three components written in this column a to b3 minus a3 b2 a1 b3 minus a3 b1 a1 b2 minus a2 p1 ok the three components written in this column this is something that you have seen but you know if we can also write it as a matrix times the vector B so we have this column of B b1 b2 b3 and a matrix it&#39;s very special matrix which has 0 on the diagonal and the off diagonal components are negative of each other so this is basically a skew symmetric matrix the components of the matrix are formed by the components of a okay for example this first row second column one two first row second column is formed by the third component of a so we have III here first row third column is found by second component of a so whatever is missing you take that component first row third column second is missing so it&#39;s formed by second component e to second row third column one is missing so it&#39;s found by first component of a II 1 and then the tricky thing is this first one a 3 has to be with a minus sign a 2 remains a 2 and a 1 becomes minus a 1 and then you form this 2 symmetric matrix okay so this is a 3 here we have minus a 2 and here we have a 1 alright so there is a nice way to form this matrix and when you work out this matrix times the column you will indeed see what you have on the left hand side ok the cross product and this matrix so people say that the column of e is actually the axial you know just named axial of this is Q symmetric matrix so whenever you have its two symmetric matrix you can form a column from the three entries in that excuse metric matrix or whenever you have a column you can form it skew symmetric matrix in the way I just told you all right so this is a different way to visualize cross product ok and what you have to remember also that just like the vector which is independent of the coordinate system these operations dot product or cross-product are also independent of the coordinate system for example a dot B is summation of a ibi if you use different coordinate system then it would be summation of AI hat bi hat but the product with the summation has to be the same so dot product will not change likewise for the cross product also so they are geometric they have geometric meaning what is dot product you have two vectors then the dot product is take the magnitude of each of the vectors multiplied with the cause of the angle between the two vectors so it comes from geometry you change the coordinate system magnitude of the vectors do not change the angle between the vectors do not change so the dot product will also not change right so dot product and cross product are basically coordinate free operations when you write it you write it in terms of the components of the vector with change but as a whole it has to be the same there is another interesting operations that one can do with vectors you see dot product gave you a scalar cross product gives you vector this another operation which is called tensor product a tensor product B so the tensor product is denoted by this cross symbol with a circle alright so this gives you what&#39;s called a second order tensor ok so this is the first time we are learning about tensor so this gives you a second order tensor so it&#39;s second order so we put two tildes below ok second order tensor the way to see this easily is we say whenever we have tensor product and you try to write it in the column form then you take the first vector as a column but the second vector as a row incorrectly reverse of what we did with the dot product in dot product the first one was a row right but here the second one is row so since the second one is a row so therefore it becomes the transpose isn&#39;t it it&#39;s the transpose of the column first one is 3 cross 1 3 rules 1 column the second one is now 1 Cross 3 so when you do the multiplication what you get is actually a matrix isn&#39;t it it&#39;s a matrix and the individual components are denoted using two subscripts C IJ okay so C IJ for example the eyath row and jth column what is C IJ when you work this out it is simply a I into B G so you see for the vectors the components have single index subscript whereas for C you have two subscripts there were their first called second order tensor okay and so we use two tilde&#39;s 1 tilde for the vector two tildes for second order tensor 3 T loves for third order tensor all right you know we we can also see it that the see was a tense a bit of B right now why don&#39;t I expand a as summation over I AI e I ok the vector e and this has to be tensor product with the vector B so that summation over j bj ej correct now we could write the same thing as double summation IJ AI BJ e I tensor product EJ now contrast this with the vector representation for a vector we say a is equal to summation over I AI e I so we have I going from 1 2 3 because it has three basis vectors e1 e2 e3 for second-order tensors the basis vectors are these quantities AI tensor product eg not just II 1 e 2 e 3 with a tensor product so how many of the basis vectors are there there are nine of them I equal to 1 to 3 J equal to 1 to 3 so there are nine basis vectors okay so basis tensors so there are nine basis tenses so any tensor you could write it in terms of these nine basis tensors if you change the coordinate system from e1 e2 e3 to even hat e 2 hat III hat then your basis chances are a I hat tensor product EJ hat okay and the component with respect to death basis is CIJ alright so here this AI into bj is CI j and for a general second order tensor so we write that c is equal to summation over IJ CIJ e I tensor product EJ and C IJ is then the component of D tensor C with respect to the basis tensor AI tensor product ej okay let us see what we get it in the matrix form so this double summation take one of the terms from there let us say c 1 2 e 1 tends to product e 2 now if I write it in the column form the first one will be a column where the second one becomes a row now the column form of e 1 in the e 1 e 2 e 3 coordinate system what will that be that&#39;s simply 1 0 0 right because component of e 1 along even is 1 along other directions is 0 so even becomes 1 0 0 whereas e 2 becomes 0 1 0 remember these are the representations of e 1 and E 2 in E 1 e 2 e 3 : system itself if you use some other quarter system to represent II 1 you will get something else you won&#39;t get 1 0 0 you have to first find the components in the new coordinate system and then write those components in the column ok anyway so in this coordinate system now you work out this multiplication and what you get you get a matrix 3 by 3 where only the first row second column is nonzero and that is c1 2 rest all are 0 all right so similarly you have several other terms in this representation of tensor and now you can see where is CIJ going so second order tensor when you represent it in a coordinate system you get a matrix okay for vectors you get a column but for tensors you get a matrix and the coefficient with respect to the basis vectors they form the components of the matrix okay and CIJ goes in the I ate row and jth column right you can see it here so I just write here all right and the see with this big bracket so this is the matrix form of the tensor C and the matrix form will change from one coordinate system to the other coordinate system but the tensor itself will not change okay so you could write this same tensor C as double summation IJ CIJ e I tensor product EJ or as double summation IJ CIJ hat of AI hat tensor product EJ hat so two different ways to write the same tensor because we have used two different coordinate system so from here the matrix form has C IJ right whereas from here the matrix form has got C IJ hat so we get two different matrices but the tensor itself is the same okay and just to give a recap the vector with a single tilde then we say it is a first order tensor okay whereas scalars without any tilde it is a scalar and then we say it is a zero its order tensor you put two tilde it&#39;s second order tensor you put three tilde it becomes a third order tensor okay and you can keep on thinking of higher order tensors also alright the next thing is how does a second order tensor multiplies with a vector okay alright so suppose we say a is equal to C times B okay so C is the second order tensor B the vector I want to multiply them and I want to see what do I get so I&#39;m going to write the full form of C and B and let&#39;s see how do they multiply so the first one see has got double summation over IJ CIJ e I tensor product eg the second one has got a single summation say over K of b k èk and now we have to define this multiplication so we will say that the second row tensor and the first row tensor when they multiply then we take the second vector from the tensor so EJ and ek and take their dot product so that&#39;s how we define the multiplication you don&#39;t take E I and E K you have to take EJ and EJ take the dot product and let&#39;s see what we get so two vectors you take dot product it becomes a scalar right so here then we have triple summation right we have three i j k CI j BK so all these scalar quantities you first write them together and then we have the vector quantities so we have e I tensor product EJ into e K and we know we have to take a dot product of EJ and EJ so this becomes see i j BK ii i of ej dot each a because that&#39;s the definition of multiplication so what you get when you dot EG and EK e 1 dot e 1 what is it 1 e 1 dot e to the 0 right towards the perpendicular so when J and K are same it is 1 otherwise it is 0 so there is a very nice function in mathematics which is called Kronecker Delta function and we write it as Delta IJ delta subscript IJ and we say it is equal to 1 if i equal to j and is equal to 0 if i not equal to j ok so we can use this kronecker delta here and it becomes triple summation CI j BK e i delta JK and now you have to see as for the definition if j and k are not the same you get 0 so think of a summation over k so you have k equal to 1 2 3 so that summation will give you a nonzero quantity only when k is equal to j right so basically we can actually get rid of the summation over k and wherever we have k we replace it with j okay so this then becomes summation over IJ CIJ be J CK has been replaced with J times e I and that&#39;s your vector a right to begin with we said a is equal to C times B correct and you see carefully if you put your bracket here isn&#39;t this a I because vector a you could write it as AI e I so this is basically a ie so what we see is that when we multiply a second-order tensor with a vector we get a vector and the components of the vector are given by this formula so e AI is equal to summation over j CI j bj right if you try to write this in a column form then this is basically the i yet row of C matrix multiplied with the column of b c b1 b2 b3 c i 1c i to see i 3 you multiply it you get what you have here so this actually implies if you want to get the whole column of a a1 a2 a3 then it is simply the matrix c times the column of b so what we have found out finally is that if you want to multiply a second row tensor with a vector the resulting is a vector and to get the column of the resulting vector you simply multiply these C matrix with the column of being usual matrix vector multiplication so I just write here you know with this in hand we can actually go back to our cross product definition do you recall when we had a cross PE then we said in the column form a cross B was actually the matrix times the column of B isn&#39;t it so now in the tensor form we can say that the resulting vector C which is a cross B could also be written as second order tensor a times B because second or tensor times vector is equivalent to matrix vector multiplication and now the second order tensor a its column is is this excuse metric matrix a ok and we say that this is a skew symmetric tensor okay I hope all of you are with me so we have to always differentiate between tensor and its representation representation is the matrix which changes from one correlation to the other whereas tensor is independent of the coolant system the other thing is for a vector we know that VI is equal to V dot E I isn&#39;t it for a tensor if I let&#39;s say I want to extract the coefficient of the tensor what should I do so ckl for example what is its formula okay so we are going to work that out so I will prove you that see KL is equal actually C times e l dot with EK can you see how could you get that formula suppose I want c12 okay let&#39;s see for C 1 2 so C 1 2 means C times e 2 dot with E 1 so this is right now a tensor representation let&#39;s think of writing this in the matrix form okay so C becomes the C matrix e 2 becomes the column of e 2 and E 1 becomes the column of e 1 so this is same as C matrix the whole C matrix c11 c12 c-13 the whole C matrix multiplied by e 2 and e 2 is then 0 1 0 right you multiply these two you get a vector or a column and then I dot it with E 1 which is 1 0 0 so can you see what you will get within that small bracket you have matrix times this vector 0 1 0 when you multiply you simply get the second column of C matrix ok so you get c1 to c2 to c3 to the second column and then dot with 1 0 0 and you get c1 2 right if you want to extract c1 to this C times e 2 dot with E 1 ok that&#39;s the way you can extract the component of a tensor relative to its basis tensor all right similarly you can also show when you multiply two tensors you know multiplying two second-order tensors so there are actually various forms of multiplication one is where you multiply two second or tensors and you get the second order tensor itself a different second order tensor okay so suppose we have C equal to a into B so now we have to see how does this multiplication work out okay so we can expand the whole thing I&#39;m in the right hand side and let&#39;s work see what we get so E is then summation over IJ a IJ e I tensor product a J times B which is again double summation over let&#39;s say KL of BK l EK tensor product e l right so now let&#39;s bring all the scalar things together and the vectors on the other side so we have four summations i g ki l AI j BK l right ii I tensor product EJ multiplied with ek tensor product e L so this multiplication has to be done such that the inner vectors that you have you take the dot product of the inner vectors dot product of the inner vectors which are EJ and EJ here and let&#39;s see what we get so you simply have Delta JK right so you have AI tensor product e l CJ and ki are gone and only ein e l are present in to Delta JK now how to work this further ahead so we have two summations one over J + 1 over K and then we have Delta JK so you know what to do right Delta JK is equal to 1 only when J is equal to K so there are two ways to proceed either you remove the summation over J and replace J with K everywhere or you remove the summation over K and replace K with J right just like we were doing earlier but now we have two ways to go ahead because we have summation over both J and K so let us suppose we remove summation over J and replace J with K so we have triple summation i qi l ii i k because J gets replaced with k be k l ii i tensor product e l correct and we can write the same thing as summation over I L and then a summation of a key of AI K BK l e I tensor product e L okay and you see this has to be see the tense of C so don&#39;t you think the thing inside the bracket is actually C I and isn&#39;t it so this thing inside the bracket is nothing but C I L so now we know what happens when two tensors multiply what happens to its matrix form so CIL is simply summation over K of aik into BK l and can you see what this is really this is actually we take the IH row of a ei1 a I to a i3 so that&#39;s the I it row of a matrix multiplied with LS column of B isn&#39;t it to get CIL you take the is row of a multiply with LS column of B so to get the full matrix C you then simply have to do a matrix into B matrix so this implies that your C matrix is simply a matrix times the B matrix a very simple formula finally so this means when two tensors are multiplying then the multiplication that we worked out is as if you just multiply the matrix form of the two tensors in the usual way and then the product matrix that you get is the matrix form of the resulting tensor okay very simple so this is about tensors now there will many more things as we proceed in the lecture some new things we may have to learn so that I will address you as we work out the lecture ok there is also something called rotation tensor so this is the physical rotation you have a vector you rotate you get a new vector so we are thinking it&#39;s physically but mathematically what should you do on that vector to get the new rotated vector so this rotation is actually done through the rotation tensor okay so you can think of a set of three perpendicular vectors e1 e2 e3 and then it gets rotated into even hat e 2 hat and III hat okay for this rotation there is always a unique rotation tensor so you have a set of tried three vectors and another set of three vectors you can always find a rotation so that it Maps you into the new set of three vectors okay from the first set of three vectors you multiply and you get the second set of three vectors there is a unique rotation to do this operation so we will say that E I hat is equal to R times e I for each of the I equal to 1 2 3 all right and this rotation has got a unique property this rotation tensor it turns out its matrix form is orthogonal matrix okay so are we say it&#39;s a Aalto gonal tensor and its matrix form is a orthogonal matrix and such matrices have unique property that is you take R multiplied with its transpose or you take R transpose and multiplied with r then you get identity an identity matrix what does this mean so if we look at the matrix form then we have our matrix multiplied with the transpose of the matrix giving you identity right one one zero zero zero zero zero zero suppose I call this matrix C then what is C IJ so C IJ is nothing but Delta IJ right it&#39;s equal to one when I is equal to J that means the diagonal is one off diagonal is zero but you also know from the matrix multiplication you have R into R transpose so we just worked out CIJ when you have two matrices multiplying the resulting matrix coefficient C I L for example is aik bkl right so what will be CIJ so it will be a summation over sum K are I K into R transpose k j the inner index is summed you see the inner index inner index is summed but then our transpose kg is same as our JK isn&#39;t it what do we do in transpose we just flip the off diagonal components right so this is same as summation over k RI ki r JK which is equal to Delta IJ can you see its meaning so this says you take the iith rho of r + j throw our and you take the dot product right dot of i a throw and jaea throw that&#39;s equal to delta IJ so if you take two different rows of this r matrix and U dot it you get 0 if you take a row and dot it with itself you get 1 so this tells us that the rows of this R matrix are perpendicular to each other similarly you can also prove that the columns of the r matrix are also perpendicular to each other so this rotation matrix is very unique than the rows and columns of this matrix so I&#39;ll say two rows any two rows and same thing goes for columns or this matrix are we say orthonormal to each other so basically if the two rows are different it is zero if the two rows are same it is 1 so that means ortho normal ortho for perpendicular normal for 1 normalized okay so that&#39;s about the rotation tensor let&#39;s see an example of how will the rotation matrix look like so we have this e 1 e 2 e 3 coordinate system and suppose I rotate this coordinate system and think of a new coordinate system even had eetu hat and III hat where III hat is same as III but even hat and E 1 are rotated by theta okay so how will these rotors and matrix look like that&#39;s the question here what is this R equal to the components of our now do you recall for a tensor what is R IJ you can go back you see ckl is C times e l dot e K so R IJ is then rotation tensor R into e j dot with E AI but by definition of this rotation tensor are into EJ is nothing but EJ hat dot with E I correct so do you see now what is our I J so the J&#39;s column of this rotation matrix is the EJ hat vector written in the even e 2 e 3 corner system can you see it so I have this rotation matrix let&#39;s say J is equal to 2 so we have R 1 to R 2 to R 3 2 so our I 2 is simply e 2 hat dot e I so R 1 2 for example is e 2 hat dot e 1 the component of e 2 hat along even R 2 2 is e 2 hat dot e to the component of e 2 hat along e 2 likewise for R 3 2 so this column is basically e 2 hat itself but written in II 1 e 2 e 3 coordinate system that&#39;s the column e 2 hat written in even e 2 e 3 coordinate system so for this example can you guess what is e 2 hat from the picture that you see can you guess it so this angle is also theta so when you say e to hat dot e to you get cos theta when you say e to hat dot e 1 you get minus sine theta right it goes in the backward direction so what is then R 1 to e 2 hat dot e 1 that&#39;s minus sine theta R 2 to that e 2 hat dot e 2 that&#39;s cos theta and r 3 to eat uu hat dot e 3 that&#39;s 0 right similarly you can think of the first column of our matrix the first column of our matrix is the column form of e 1 hat in the e 1 e 2 e 3 coded system and the third column of our matrix is the column form of e 3 hat in the e 1 e 2 e 3 colon system so when you work whole everything out we&#39;ll finally see that the rotation matrix is simply cos theta minus sine theta 0 sine theta cos theta 0 0 0 1 let&#39;s see the third column is simply 0 you know 1 because III hat is same as e 3 ok so that&#39;s about vectors tensors rotation and in the next lecture which is actually going to be the first as far as the topic is concerned we will start with the traction vector ok thank you [Music] [Music]

Transcript for:Understanding Vectors and Tensors

Transcript for:
Understanding Vectors and Tensors