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Chapter 12: Q. 11 (page 988)

Describe the meanings of each of the following mathematical expressions:
$\nabla f (x, y, z)$

Short Answer

Expert verified

Ans: $\nabla f (x, y, z)$ It means gradient of function $f$ with a three dimensional input and a three dimensional output.

It is nicely visualized with a vector field in three dimensional space.

Step by step solution

01

Step 1. Given information.

given, $\nabla f (x, y, z)$

02

Step 2. The meaning of the given expression is,

$\nabla f (x, y, z)$ It means a gradient of function $f$ with a three-dimensional input and a three-dimensional output.

It is nicely visualized with a vector field in three-dimensional space.

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Most popular questions from this chapter

In Exercises, by considering the function $f (x, y) = x^{2} y$ subject to the constraint $x + y = 0,$ you will explore a situation in which the method of Lagrange multipliers does not provide an extremum of a function.

Explain why $(0, 0)$ is not an extremum of $f (x, y) = x^{2} y$ subject to the constraint $x + y = 0 .$

$Extrema : Find the local maxima, local minima, and saddle points of the given functions f (x, y) = x^{3} - 12 xy - y^{3} .$

Fill in the blanks to complete the limit rules. You may assume that $lim_{x \to a} f (x)$ and $lim_{x \to a} g (x)$ exists and that k is a scalar.

$lim_{x \to a} (f (x) + g (x)) =$

Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.

$\frac{\partial z}{\partial s} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

$f (x, y, z) = x y z when x^{2} + y^{2} + z^{2} = 9$

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