Let r = x i + y j + z k and r = |r|. if f = r/r p, find div f. (enter your answer in terms of r and p.) div f =

The answer in terms of r and p is div F = (3 - p)/r^p.

Step-by-step explanation:

Given:

F = R/r^p

F= <x, y, z> / |<x, y, z>|^p

F = <x/(x^2+y^2+z^2)^(p/2), y/(x^2+y^2+z^2)^(p/2), z/(x^2+y^2+z^2)^(p/2)>.

Hence,  For div F ,

We take partial derivative:

div F  = (∂/∂x) x/(x^2+y^2+z^2)^(p/2) + (∂/∂y) y/(x^2+y^2+z^2)^(p/2) + (∂/∂z) z/(x^2+y^2+z^2)^(p/2)

Now, we use the rational derivative rule to find the derivatives:

div F  = [1(x^2+y^2+z^2)^(p/2) - x * px(x^2+y^2+z^2)^(p/2 - 1)] / (x^2+y^2+z^2)^p  + [1(x^2+y^2+z^2)^(p/2) - y * py(x^2+y^2+z^2)^(p/2 - 1)] / (x^2+y^2+z^2)^p  + [1(x^2+y^2+z^2)^(p/2) - z * pz(x^2+y^2+z^2)^(p/2 - 1)] / (x^2+y^2+z^2)^p

div F  = (x^2+y^2+z^2)^(p/2 - 1) {[(x^2+y^2+z^2) - px^2] + [(x^2+y^2+z^2) - py^2] + [(x^2+y^2+z^2) - pz^2]}  / (x^2+y^2+z^2)^p

div F  = [3(x^2+y^2+z^2 - p(x^2+y^2+z^2)] / (x^2+y^2+z^2)^(p/2 + 1)

div F  = (3 - p) (x^2+y^2+z^2) / (x^2+y^2+z^2)^(p/2 + 1)

div F  = (3 - p)/(x^2+y^2+z^2)^(p/2)

Now it comes like,

div F  = (3 - p)/r^p.

47 - 90 = 43

43 - 180 = x

x = 137

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: