MTH 201 Week 3: Gradient Vectors and Their Applications

Introduction

• Last week: Multivariable functions and partial derivatives
• This week: Gradient vectors, directional derivatives, and optimization

Gradient Vector Definition

• Gradient Vector: Collection of first-order partial derivatives
• Represented using the nabla (∇) operator
  • For function ( f(x, y) ), the gradient ( \nabla f = (f_x, f_y) )
  • ( f_x ) and ( f_y ) are partial derivatives with respect to ( x ) and ( y ) respectively
• Can be extended to functions with more than two inputs

Example: Calculating the Gradient

• Given function: Expand middle term to simplify calculations
• Partial Derivatives:
  • With respect to ( x ): Only terms with ( x ) contribute
  • With respect to ( y ): Only terms with ( y ) contribute
• Collect partial derivatives to form the gradient vector

Application of Gradient Vectors

• Visualization: Use contour plot to visualize gradient vectors
• Example Point: ( x = 0, y = -1.5 )
  • Gradient calculation: ( \nabla f = (0, 3) )
  • Indicates direction of steepest ascent at specific point
  • Vector points upwards along the y-axis

Interpretation of Gradient Vectors

• Gradient Direction:
  • Points in the direction of the maximum rate of change of the function ( f )
  • Represents the direction of steepest ascent
• Contour Plot Analysis:
  • Arrows on the plot indicate direction of gradient vectors
  • Gradient vectors point towards increasing function values
• Surface Plot Comparison:
  • Visualize steepest ascent direction on surface plot

Conclusion

• Calculating gradient vectors allows determination of direction for maximum function increase
• Useful for optimizing functions and determining directional derivatives

This week's lecture provides a foundational understanding of how gradient vectors function in multivariable calculus and lays the groundwork for further exploration into optimization techniques.