https://file.aiquickdraw.com/mathbot/file/1726376743383ej5zfpgm.jpg

To find the gradient of the scalar point function \( f(x, y, z) = x^2 + y^2 + z^2 \), we need to compute the partial derivatives of \( f \) with respect to each of the variables \( x \), \( y \), and \( z \).

1. The partial derivative of \( f(x, y, z) \) with respect to \( x \):
   \[
   \frac{\partial f}{\partial x} = 2x
   \]

2. The partial derivative of \( f(x, y, z) \) with respect to \( y \):
   \[
   \frac{\partial f}{\partial y} = 2y
   \]

3. The partial derivative of \( f(x, y, z) \) with respect to \( z \):
   \[
   \frac{\partial f}{\partial z} = 2z
   \]

Therefore, the gradient of \( f(x, y, z) \) is the vector formed by these partial derivatives:
\[
\nabla f(x, y, z) = (2x, 2y, 2z)
\]

Looking at the options, the correct answer is \( \boxed{(2x, 2y, 2z)} \).

Would you like a deeper explanation on gradients or any of the steps? Here are five related questions:

1. How do you compute the gradient of a more complex function?
2. What is the physical meaning of the gradient in the context of a scalar field?
3. How can the gradient be used to find the direction of steepest ascent?
4. How do you find the gradient for vector fields?
5. What is the divergence and how does it relate to the gradient?

**Tip:** The gradient points in the direction of the greatest rate of increase of a function, and its magnitude represents the rate of increase.

The gradient of the scalar point function f(x,y,z)=x^2+y^2+z^2 is?

Find the gradient of the scalar point function f(x, y, z) = x^2 + y^2 + z^2 using partial derivatives. This problem explains how to compute the gradient for multivariable functions, suitable for Grades 11-12 and college-level students studying vector calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation