Calculus III Lecture 16: Directional Derivatives and the Gradient Vector

Review of Partial Derivatives

• Partial derivatives with respect to X or Y indicate the rate of change along the x-axis or y-axis, respectively.
• Example: Temperature on a flat plate can be expressed as a function of x and y.
  • Partial w.r.t X: Rate of change of temperature in the horizontal direction.
  • Partial w.r.t Y: Rate of change of temperature in the vertical direction.

Directional Derivatives

• We can move in any direction, not just along x or y axes, to find the rate of change using directional derivatives.
• Direction described by unit vectors. For example, unit vector u with components a and b.
• Process:
  1. Draw a line containing vector u (line l).
  2. Locate points P (where we study rate of change) and Q (another point on line l).
  3. PQ = H * u (H is a scaler).
  4. Use coordinates of P to find coordinates of Q.
  5. Calculate the average rate of change as we move from P to Q: (Delta T)/H.
  6. Limit as H approaches 0 gives the directional derivative.
• Formula: Directional derivative of f at (x0, y0) in direction u (with components a, b) = limit as H approaches 0 [(f(x0+ha, y0+hb) - f(x0, y0)) / H].

Simplified Theorem for Directional Derivatives

• If f is differentiable, the directional derivative in direction (a, b) is:
  • D_u f(x0, y0) = (f_x * a) + (f_y * b).
• This simplifies to using partial derivatives and unit vector components.

Gradient Vector

• The gradient of a function f is denoted by an upside-down triangle (∇f) and is composed of partial derivatives with respect to each variable.
  • ∇f = (f_x, f_y) for functions of two variables.
• Gradient in more depth studies how the directional derivative value changes with the direction u.
  • Relationship: D_u f(x0, y0) = ∇f · u = |∇f| |u| cos(θ), where θ is the angle between the gradient vector and u.
  • Cosine values indicate maximum (1, parallel), minimum (-1, anti-parallel), and zero (perpendicular).
  • Conclusion: Maximum value of directional derivative is |∇f|, achieved when u is in the direction of ∇f.

Application Examples

1. Find directional derivative of f(x, y) = x/y at (6, -2) in the direction of vector V (-1, 3).
   • Convert V to unit vector.
   • Compute partial derivatives with respect to x and y.
   • Use directional derivative theorem: D_u f(x0, y0).
2. Given a function, find the direction u such that the directional derivative has a given value.
   • Solve using partial derivatives and unit vector properties.

Gradient Vector and Tangent Planes

• The gradient vector is orthogonal to the tangent plane of a surface at a given point.
• Formula for the tangent plane to a surface f(x, y, z) = k:
  • Use gradient (∇f) evaluated at the given point P as the normal vector.
  • Standard form: ax + by + cz = d.
• Normal line to the surface can also be described using gradient vector as the direction vector.

Summary of Gradient Uses

• Normal vector for tangent plane.
• Direction vector for normal line.
• Direction for greatest rate of increase and its magnitude.
• Opposite gradient gives the direction for the greatest rate of decrease.
• Orthogonal to level curves and surfaces.
• Useful in finding maximum value of directional derivatives.

Example Problems

• Walk due south/northwest: Determine whether ascension or descension happens and the rate, using gradient vector.
• Finding points on a surface where the tangent plane is parallel to a given plane.

Conclusion: The gradient vector is a powerful tool in calculus for solving problems related to directional derivatives, maximum rate of change, tangent planes, and much more.