CALCU- Find p from p= F in the case where\[F=(a x

\therefore

Find

p

from

\boldsymbol{\nabla} p=\rho \mathbf{F}

in the case where

\mathbf{F}=(a x-b y, a y+b x, a) \text { and } \rho=\left(x^{2}+y^{2}\right)^{-z},

with constants

a, b

.

Solution by Steps

step 1

To find

p

from

\boldsymbol{\nabla} p = \rho \mathbf{F}

, we need to solve the equation given the vector field

\mathbf{F}

and the scalar field

\rho

step 2

Given

\mathbf{F} = (ax - by, ay + bx, a)

and

\rho = (x^2 + y^2)^{-z}

, we can equate

\boldsymbol{\nabla} p

to

\rho \mathbf{F}

step 3

Solving the equation

\frac{\pi}{200} p = (x^2 + y^2)^{-z} (ax - by, ay + bx, a)

for

p

, we get different cases for the solution

step 4

For the case where

b = -a

and

x = 0

and

y = 1

, we find

p = \frac{200a}{\pi}

step 5

For the case where

x \neq 0

and

(x^2 + \frac{1}{4}(1 - \sqrt{-4x^2 + 4x + 1})^2)^z \neq 0

, we find

p = \frac{200a(x^2 + \frac{1}{4}(1 - \sqrt{-4x^2 + 4x + 1})^2)^{-z}}{\pi}

and

b = \frac{a(\sqrt{-4x^2 + 4x + 1} + 1)}{2x}

and

y = \frac{1}{2}(1 - \sqrt{-4x^2 + 4x + 1})

step 6

For the case where

x \neq 0

and

(x^2 + \frac{1}{4}(\sqrt{-4x^2 + 4x + 1} + 1)^2)^z \neq 0

, we find

p = \frac{200a(x^2 + \frac{1}{4}(\sqrt{-4x^2 + 4x + 1} + 1)^2)^{-z}}{\pi}

and

b = -\frac{a(\sqrt{-4x^2 + 4x + 1} - 1)}{2x}

and

y = \frac{1}{2}(\sqrt{-4x^2 + 4x + 1} + 1)

step 7

For the case where

p = 0

and

(y - 1)y(y^2)^z \neq 0

, we find

a = 0

and

b = 0

and

x = 0

Answer

The value of

p

depends on the specific case and values of

a

,

b

,

x

,

y

, and

z

. The solutions are given by the cases in steps 4 to 7.

Key Concept

Gradient and Scalar Field Multiplication

Explanation

The key concept involves solving for a scalar field

p

when its gradient is given by the product of another scalar field

\rho

and a vector field

\mathbf{F}

. The solution requires equating the gradient of

p

to this product and solving for

p

under different conditions for the variables involved.

算出答案

答案：

p

的值取决于特定情况和

a

，

b

，

x

，

y

和

z

的值。解决方案由步骤4到7中的情况给出。

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