Multivariable Calculus Lecture Notes

Introduction

• Lecture series on multivariable calculus.

Recap of Previous Lecture

• Defined limits for several variable functions.
• Limit notation:
  [ \lim_{(x,y) \to (x_0,y_0)} f(x,y) = l ]
  Means: For every ( \epsilon > 0 ), there exists ( \delta > 0 ) such that:
  [ |f(x,y) - l| < \epsilon \text{ whenever } 0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta ]
• Geometric interpretation of limits in multivariable functions.

Important Properties of Limits

1. Uniqueness: If the limit exists, it is unique.
2. Conversion to Polar Coordinates:
   • Substitute:
     • ( x - x_0 = r \cos \theta )
     • ( y - y_0 = r \sin \theta )
   • Then:
     [ r^2 = (x - x_0)^2 + (y - y_0)^2 ]
   • As ( r \to 0 ), ( (x,y) \to (x_0,y_0) )
3. Delta-Epsilon Definition:
   • For every ( \epsilon > 0 ), choose ( \delta < \epsilon ) to ensure limit properties hold.

Example Problem

• Limit to Prove: ( \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2} = 0 )
  • Using Polar Coordinates:
    • Substitute ( x = r \cos \theta ), ( y = r \sin \theta ).
    • Resulting limit: ( \lim_{r \to 0} r \cos^3 \theta = 0 ).
  • Using Cartesian Coordinates:
    • Show that ( |x| < \delta \Rightarrow |x^3| < |x| ).
    • Choose ( \delta = \epsilon ) to satisfy limit.

Two Path Test

• Non-existence of Limit: If limits from two different paths yield different results, then the limit does not exist.
• Example:
  • Limit ( \lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2} ) along different paths results in different values, indicating non-existence.

Iterated Limits vs. Double Limits

• Double Limits: ( \lim_{(x,y) \to (x_0,y_0)} f(x,y) = l )
• Iterated Limits:
  • ( \lim_{x \to x_0} \lim_{y \to y_0} f(x,y) )
  • Exists if:
    1. Both iterated limits are equal.
    2. Both limits must exist.
• Example of Non-existence:
  • If iterated limits yield different values, then the double limit does not exist.
  • Use paths like ( y = mx ) to demonstrate path dependency.

Conclusion

• To establish limit existence: Use delta-epsilon definition.
• To prove limit does not exist: Demonstrate path dependency or different values from multiple paths.

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Notes Summary

• Key concepts from the lecture on limits in multivariable calculus.
• Includes definitions, properties, examples, and tests for limit existence.