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Chapter 6: Q20P (page 295)

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent 鈭�r2

Short Answer

Expert verified

鈭�r2=2rer

Step by step solution

01

Given Information

The given information is 鈭�r2 and r=x2+y2.

02

Definition of Scalar field.

In mathematics and physics, scalar field or scalar-valued function referred to a scalarvalue to everypointin aspace鈥� possiblyphysical space. The scalar may either be a (dimensionless)mathematical numberor aphysical quantity.

03

Finding the gradient

Using the equation,

鈭�蠒=grad蠒=i鈭�蠒鈭�x+j鈭�蠒鈭�y+k鈭�蠒鈭�z

Find the value of gradient.

r2=x2+y2鈭�r2=2xi^+2yj^

鈭�r2=2x脳x2+y2x2+y2i^+2y脳x2+y2x2+y2j^=2x2+y2xx2+y2i^+yx2+y2j^=2rcos胃i^+sin胃j^=2rer

Hence, the solution to this problem is鈭�r2=2rer .

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Most popular questions from this chapter

鈭�(yi鈭�xj+zk)鈭�draround the circumference of the circle of radius 2, center at the origin, in the (x,y)plane.

鈭�2xdy-3ydxaround the square with vertices(0,2),(2,0),(-2,0),(0,-2).

Find vector fields A such that V=curlAfor each given V.

V=-K

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent 鈭�x.

Verify that the force field is conservative. Then find a scalar potential 胃 such that F=-鈭�蠁,F=z2sinhyj+2zcoshyk,k=constant.
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