Question

To compute the gradient vector fields of the given functions, you need to find the partial derivatives of the functions with respect to both x and y, and then construct the gradient vector using these partial derivatives. Let's compute them:

A. For f(x,y)=5x² +9y2

Answer 1

To calculate the gradient vector field of f(x,y) = 5x² + 9y², determine the partial derivatives 10x and 18y with respect to x and y, respectively. Assemble these into the gradient vector as grad(f) = (10x, 18y), which indicates the direction of steepest ascent for function f.

Step-by-step explanation:

The task involves computing the gradient vector field of the function f(x,y) = 5x² + 9y². To find the gradient, we need the partial derivatives of the function with respect to both x and y. The partial derivative with respect to x is ∂f/∂x = 10x, and the partial derivative with respect to y is ∂f/∂y = 18y. The gradient vector is thus constructed as grad(f) = ∇f = (10x, 18y). This represents the direction and rate of the steepest ascent of the function f at any point (x,y).

Partial derivatives are calculated by treating all other variables as constants besides the one being derived. In this two-dimensional scenario, the z-component is not present, but in three dimensions, it would be included. When considering physical systems, the force relation to potential energy is represented as the negative gradient vector of the potential energy function, although that's beyond the current two-dimensional scope.