Chapter 12: Q. 2 (page 952)

What is the definition of the directional derivative for a function of three variables, $f (x, y, z)$ ? Be sure to include the words "unit vector" in your definition.

Short Answer

Expert verified

Going to assume that limit exists is $D_{u} f (x_{0}, y_{0}, z_{0}) = {Lim}_{h \to 0} \frac{f (x_{0} + a \cdot h, y_{0} + b \cdot h, z_{0} + c \cdot h) - f (x_{0}, y_{0}, z_{0})}{h}$

Step by step solution

Unit vector.

A vector could be a quantity that has both a magnitude and a directionrelated to it.
A unit vector could be a vector with a magnitude of $1$
It's alsoobserved as a Direction Vector.

Directional Derivative

Directional Derivative of a function of three variables:

Let $f (x, y, z)$ be a function of two variables defined on an open set containing the point $(x_{0}, y_{0}, z_{0})$ , and let $u = (a, b, c)$ be a unit vector.

The directional derivative with in $f$ at $(x_{o,} y_{o,} z_{o})$ in the direction of $u$ , denoted by

localid="1650479588198" $D_{u} f (x_{0}, y_{0}, z_{o})$ , is given by;

$D_{u} f (x_{0}, y_{0}, z_{0}) = {Lim}_{h \to 0} \frac{f (x_{0} + a \cdot h, y_{0} + b \cdot h, z_{0} + c \cdot h) - f (x_{0}, y_{0}, z_{0})}{h}$

Provided that a limit existed

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What is the definition of the directional derivative for a function of three variables, $f (x, y, z)$ ? Be sure to include the words "unit vector" in your definition.

Short Answer

Step by step solution

Unit vector.

Directional Derivative

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What is the definition of the directional derivative for a function of three variables, f(x,y,z) ? Be sure to include the words "unit vector" in your definition.

Short Answer

Step by step solution

Unit vector.

Directional Derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Mechanics Maths

Logic and Functions

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

What is the definition of the directional derivative for a function of three variables, $f (x, y, z)$ ? Be sure to include the words "unit vector" in your definition.