Vector-field-F-R-3-R-3-F-x-y-z-xye-1-5ye-2-3yze-3-S-x-2-y-2-z-2-r-2-F-dS-

Question Number 216351 by issac last updated on 05/Feb/25

Vector field \vec{F}; R^{3} \to R^{3}

\vec{F} (x, y, z) = x y {\vec{e}}_{1} - 5 y {\vec{e}}_{2} - 3 y z {\vec{e}}_{3}

\underset{S; x^{2} + y^{2} + z^{2} = r^{2}}{\int \int} \vec{F} \cdot d \vec{S} = ?

Answered by MrGaster last updated on 06/Feb/25

▽ \cdot \vec{F} = \frac{\partial}{\partial x} (\partial y) + \frac{\partial}{\partial y} (- 5 y) + \frac{\partial}{\partial z} (- 3 y z)

= y - 5 - 3 y = - 2 y - 5

\int \int \int_{V} (- 2 y - 5) d V

x = r \sin θ \cos ϕ, y = r \sin θ \sin ϕ, z = r \cos θ

d V = r^{2} \sin θ d r d θ d ϕ

0 \leq r \leq r, 0 \leq θ \leq π, 0 \leq ϕ \leq 2 π

\int \int \int_{V} (- 2 y - 5) d V = \int_{0}^{r} \int_{0}^{π} \int_{0}^{2 π} (- 2 r \sin θ ϕ - 5) r^{2} \sin θ d ϕ d θ d r

= \int_{0}^{r} \int_{0}^{π} [- 2 r^{3} \sin^{2} θ \int_{0}^{2 π} \sin ϕ d ϕ - {5 r}^{2} \sin θ \int_{0}^{2 π} d ϕ] d θ d r

\int_{0}^{r} \int_{0}^{π} [0 - 10 π r^{2} \sin θ] d θ d r

= - 10 π \int_{0}^{r} \int_{0}^{π} r^{2} \sin θ d θ d r

= - 10 π \int_{0}^{r} \int_{0}^{π} {[- r^{2} \cos θ]}_{0}^{π} d r

= - 10 π \int_{0}^{r} 2 r^{2} d r

= - 20 π {[\frac{r^{3}}{3}]}_{0}^{r}

= \begin{matrix} - \frac{20 π r^{3}}{3} \end{matrix}

Commented by issac last updated on 06/Feb/25

Thx! Mr Gaster you r my Hero : ⟩

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