   Chapter 14.6, Problem 37E

Chapter
Section
Textbook Problem

Show that the operation of taking the gradient of a function has the given property. Assume that u and v are differentiable functions of x and y and that a, b are constants.(a) ∇ (au + bv) = a ∇u + b ∇ v(b) ∇(uv) = u ∇ v + v ∇ u(c) ∇ ( u v ) = v ∇ u − v ∇ u v 2 (d) ∇un = nun−1 ∇u

(a)

To determine

To show: The equation (au+bv)=au+bv , where uandv are differentiable functions of xandy and that aandb are constants.

Explanation

Proof:

Given:

uandv are the functions of xandy .

aandb are constants.

Calculation:

Consider the equation (au+bv) .

Compute the value of (au+bv) as follows,

(au+bv)=x(au+bv),y(au+bv)=x(au)+x(bv),y(au)+

(b)

To determine

To show: The equation (uv)=uv+vu , where uandv are differentiable functions of xandy and that aandb are constants.

(c)

To determine

To show: The equation (uv)=vuuvv2 , where uandv are differentiable functions of xandy and that aandb are constants.

(d)

To determine

To show: The equation (un)=nun1u , where uandv are differentiable functions of xandy and that aandb are constants.

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