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Chapter 13: Problem 3

Interpret the direction of the gradient vector at a point.

Short Answer

Expert verified
Answer: To find the direction of the gradient vector at the point (x0, y0, z0), calculate the gradient vector 鈭噁(x, y, z) by computing its partial derivatives with respect to x, y, and z. Then, evaluate the gradient at the given point, 鈭噁(x0, y0, z0). Analyze the angles the gradient vector forms with the x, y and z axes to determine the direction in which the function increases most rapidly.

Step by step solution

01

Find the gradient of the function

Let's assume we are given a scalar function f(x, y, z). To find the gradient of the function, we need to compute its partial derivatives with respect to x, y, and z. The gradient vector is given by: \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) Calculate the partial derivatives of the function with respect to x, y, and z.
02

Evaluate the gradient at the point

Let's say we want to evaluate the gradient at the point (x0, y0, z0). To do this, plug the point's coordinates into the gradient vector: \nabla f(x0, y0, z0) = \left(\frac{\partial f}{\partial x}(x0, y0, z0), \frac{\partial f}{\partial y}(x0, y0, z0), \frac{\partial f}{\partial z}(x0, y0, z0)\right)
03

Interpret the direction of the gradient vector

The gradient vector at a point represents the direction in which the function increases most rapidly. The direction of the gradient vector is given by the angle it forms with each axis. If the gradient vector points upward in the x, y, and z-directions, it is interpreted as an increase in the function value in the positive x, positive y, and positive z directions, respectively. Conversely, if the gradient vector points downward in any of the directions, the function value decreases in that direction. With the computed gradient vector \nabla f(x0, y0, z0), analyze the angles it forms with the x, y, and z-axes to determine the direction in which the function increases most rapidly.

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