$\therefore$ 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 $a, b$.

Question

$\therefore$ 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 $a, b$.

Accepted Answer

∻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.⚹◊What is the value of $p$ in the given case of $\nabla p = \rho \mathbf{F⍭ Generate me a similar question◊

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