Surface integrals using an explicit description.

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sxbenn

12.01.2022

Parameterize by the vector function

$\vec r(x,y)=x\,\vec\imath+y\,\vec\jmath+f(x,y)\,\vec k$

so that the normal vector to is given by

$\dfrac{\partial\vec r}{\partial x}\times\dfrac{\partial\vec r}{\partial y}=\left(\vec\imath+\dfrac{\partial f}{\partial x}\,\vec k\right)\times\left(\vec\jmath+\dfrac{\partial f}{\partial y}\,\vec k\right)=-\dfrac{\partial f}{\partial x}\vec\imath-\dfrac{\partial f}{\partial y}\vec\jmath+\vec k$

with magnitude

$\left\|\dfrac{\partial\vec r}{\partial x}\times\dfrac{\partial\vec r}{\partial y}\right\|=\sqrt{\left(\dfrac{\partial f}{\partial x}\right)^2+\left(\dfrac{\partial f}{\partial y}\right)^2+1}$

In this case, the normal vector is

$\dfrac{\partial\vec r}{\partial x}\times\dfrac{\partial\vec r}{\partial y}=-\dfrac{\partial(8-x-2y)}{\partial x}\,\vec\imath-\dfrac{\partial(8-x-2y)}{\partial y}\,\vec\jmath+\vec k=\vec\imath+2\,\vec\jmath+\vec k$

with magnitude $\sqrt{1^2+2^2+1^2}=\sqrt6$ . The integral of f(x,y,z)=e^z over is then

$\displaystyle\iint_Se^z\,\mathrm d\Sigma=\sqrt6\iint_Te^{8-x-2y}\,\mathrm dy\,\mathrm dx$

where is the region in the x,y plane over which is defined. In this case, it's the triangle in the plane z=0 which we can capture with $0\le x\le8$ and $0\le y\le\frac{8-x}2$ , so that we have

$\displaystyle\sqrt6\iint_Te^{8-x-2y}\,\mathrm dx\,\mathrm dy=\sqrt6\int_0^8\int_0^{(8-x)/2}e^{8-x-2y}\,\mathrm dy\,\mathrm dx=\boxed{\sqrt{\frac32}(e^8-9)}$

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Surface integrals using an explicit description. evaluate the surface integral \iint_{s}^{}f(x,y,z)ds using an explicit representation of the surface.

f(x,y,z) = ez; s is the plane z = 8 - x - 2y in the first octant.

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Surface integrals using an explicit description. evaluate the surface integral \iint_{s}^{}f(x,y,z)ds using an explicit representation of the surface. f(x,y,z) = ez; s is the plane z = 8 - x - 2y in the first octant.

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Surface integrals using an explicit description. evaluate the surface integral \iint_{s}^{}f(x,y,z)ds using an explicit representation of the surface.

f(x,y,z) = ez; s is the plane z = 8 - x - 2y in the first octant.