Answered: Let u be a unit vector. Show that the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let u be a unit vector. Show that the directional derivative D_uf is equal to the component of V f along u.

Expert Solution

Step 1: Given Information:

Here, $u$ be a unit vector.

The directional derivative of $D_{u} f$ is equal to the component of $\nabla f$ along $u$ has to be proved.

Step 2: Calculation:

Consider a scaler valued function is $f (x, y, z)$ and the unit vector is $u = (a, b, c)$ .

The directional derivative of the function $f$ in the unit vector direction will be

$D_{u} f = \lim_{h \to 0} \frac{[f (x + a h, y + b h, z + c h) - f (x, y, z)]}{h}$ .

Consider a vector $\nabla f = (f_{x}, f_{y}, f_{z})$ and it is the gradient of the function $f$ . That is

$\begin{array}{rcl} \nabla f & = & g r a d (f) \\ = & (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}) \end{array}$

Taking the dot product of the $V$ and $u$ : then the equation will be

$\nabla f \cdot u = \frac{\partial f}{\partial x} \cdot a + \frac{\partial f}{\partial y} \cdot b + \frac{\partial f}{\partial z} \cdot c$

Here the rate of change of the function $f$ in the direction of the unit vector $u$ .

$D_{u} f = \nabla f \cdot u$

Hence, the directional derivative of the function $f$ in the direction of the unit vector $u$ is equal to the component of the gradient vector $\nabla f$ along $u$ .

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