Surface integrals

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3) Problem: Calculate the surface integral of the first kind of the function \( \vec{F} \) over the surface \( S \), where \[ \begin{array}{l} \vec{F}(x, y, z)=\left\{y^{2}-z^{2}, z^{2}-x^{2}, x^{2}-y^{2}\right\}, \\ S:\left\{\begin{array}{c} 0 \leq x \leq 4,0 \leq y \leq 4,0 \leq z \leq 4 \\ x+y+z=6 \end{array}\right. \end{array} \]

9.8.1 Surface integrals

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3 Problem: Calculate using the Ostrogradsky formula: \[ \int_{S} \int^{(x \cos \alpha+y \cos \beta+z \cos \gamma)} \underset{\sqrt{x^{2}+y^{2}+z^{2}}}{d s} \] where \( S=\left\{x^{2}+y^{2}+z^{2}=z\right\} \), \( \vec{n}=(\cos \alpha, \cos \beta, \cos \gamma) \) is the outside normal.

9.8.2 Surface integrals

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(3) Problem: Calculate the integral using Stokes formula. \[ \int_{\Gamma} x y z d x+y^{2} z d y+z x^{2} d z \] where \( \Gamma \) is the curve: \( \left\{\begin{array}{l}x^{2}+z^{2}=a^{2}, a>0 \\ y^{2}+z^{2}=a^{2}, x \geq 0\end{array}\right. \) is positively oriented on the outer side of the first cylinder.

9.8.3 Surface integrals

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3) Problem: Calculate the surface integral of the \( 1^{\text {st }} \) kind: \[ \iint_{S} \sqrt{x^{2}+y^{2}} d S \] where \( S \) is the part of the conical surface \( z=\sqrt{x^{2}+y^{2}} \), distinguished by the condition \( z \leq 1 \).

9.8.4 Surface integrals

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(3) Problem: Calculate the surface integral of the 2nd kind: \[ \iint_{S}\left(2 x^{2}+y^{2}+z^{2}\right) d y d z \] where \( S \) is the outer side of the lateral surface of the cone \( \sqrt{y^{2}+z^{2}} \leq x \leq H \).

9.8.5 Surface integrals

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(a) Problem: Calculate the surface integral of the 2nd kind: \[ \iint_{S}(5 x+y) d y d z+z d x d y \] where \( S \) is the inside of the ellipsoid \( x^{2} / 4+ \) \( +y^{2} / 9+z^{2}=1 \).

9.8.6 Surface integrals

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3) Problem: Calculate the surface integral of the \( 1^{\text {st }} \) kind: \[ \mathrm{I}=\iint_{S}\left(x^{2}+y^{2}\right) d s \] where \( S \) is the part of the conical surface \( z=\sqrt{x^{2}+y^{2}} \), distinguished by the condition \( z \leq 1 \).

9.8.7 Surface integrals

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Problem: Calculate the surface integral of the \( 2^{\text {nd }} \) kind: \[ \iint_{S} x^{3} d y d z+y^{3} d z d x \] where \( S \) is the outside of the ellipsoid part \[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1, \quad z \geq 0 . \]

9.8.8 Surface integrals

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3) Problem: Calculate the surface integral of the \( 2^{\text {nd }} \) kind: \[ \iint_{S}(5 x+y) d y d z+z d x d y \] where \( S \) is the outer side of the full surface of the cone \( x^{2}+y^{2} \leq z^{2}, 0 \leq z \leq 4 \).

9.8.9 Surface integrals

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30 Problem: Calculate the surface integral of the \( 1^{\text {st }} \) kind: \[ \iint_{\sigma}(3-2 z) d \sigma, \] where \( \sigma:\left\{\begin{array}{c}z=1-\frac{y^{2}}{2} \\ y=x, x=0, \quad z=0\end{array}\right. \)

9.8.10 Surface integrals

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(3) Problem: Find the total charge on the surface \( S \), if the surface charge density is equal to \( f(x, y, z) \) : \[ S: z=\sqrt{4-x^{2}-y^{2}}, \quad f(x, y, z)=x^{2} . \]

9.8.11 Surface integrals

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