Q. 6 - Calculus

Question

 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.

Step-by-Step Solution

Answer

$D_{u}f(x_0,y_0,z_0)=\underset{h\to 0}{\lim }\frac{f(x_0+a·h,y_0+b·h,z_0+c·h)-f(x_0,y_0,z_0)}{h}$

Step 1 : Introduction

The given is three variables $f(x,y,z)$.The objective is to find the definition of the directional derivative for a function of two variables,$f(x,y,z)$. The required definitions is provided

Step 2 :

Directional Derivative of a function of three variables:

The rate at which the function varies at a point in the direction is called the directional derivative. It can be expressed as a vector form of the ordinary derivative.

 A unit vector is a one-dimensional vector that is also known as a direction vector (Jeffreys and Jeffreys 1988).

 where represents the norm of and is the unit vector in the same direction as the (finite) vector. 

Let $f(x,y,z)$ be a function of two variables defined on an open set containing the point  and let $u=(a,b,c)$ be a unit vector. The directional derivative of  at $(x_0,y_0,z_0)$ in the direction of ,denoted by $D_{u}f(x_0,y_0,z_0)$ is given by,

$D_{u}f(x_0,y_0,z_0)=\underset{h\to 0}{\lim }\frac{f(x_0+a·h,y_0+b·h,z_0+c·h)-f(x_0,y_0,z_0)}{h}$

Provided that a limit exists.