Calculate $\int sze^z dS$ on Unit Sphere

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In summary, the integral $\displaystyle \int sze^z dS$ over the portion of the unit sphere centered at the origin with x and y less than 0 and z greater than 0 can be written as $\int_0^{\pi/2}\int_{\pi/2}^{3\pi/2} sin(\phi)cos(\phi)e^{cos(\phi)} d\theta d\phi$. The region for the integral is determined by the parametric equations of the unit sphere and the differential of surface area. The function inside the integral is $ze^z$ with $z= cos(\phi)$, resulting in the final expression.
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richatomar
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calculate $\displaystyle \int sze^z dS$
where S is the protion of the unit sphere centered at the origin such that x,y <0, z>0.
 
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richatomar said:
calculate $\displaystyle \int sze^z dS$
where S is the protion of the unit sphere centered at the origin such that x,y <0, z>0.
"Portion", not "protion"! I know that "dS" is the surface area integral but what is that small "s"?

If you mean just $\int\int ze^z dS$, then the unit sphere centered at the origin can be written as $x^2+ y^2+ z^2= 1$ or, in parametric equations, $x= cos(\theta)sin(\phi)$, $y= sin(\theta)sin(\phi)$, $z= cos(\phi)$ or, equivalently, as the vector function $(cos(\theta)sin(\phi), sin(\theta)sin(\phi), cos(\phi))$. The "differential of surface area" of the unit sphere is given by $sin(\phi)d\theta d\phi$. The region such that x and y are both negative and z is positive is that where $\theta$ is from $\pi/2$ and $3\pi/2$ and z is from 0 to $\pi/2$. $ze^z$, with $z= cos(\phi)$ is $cos(\phi)e^{cos(\phi)}$ so the integral is $\int_0^{\pi/2}\int_{\pi/2}^{3\pi/2} sin(\phi)cos(\phi)e^{cos(\phi)} d\theta d\phi$.
 

FAQ: Calculate $\int sze^z dS$ on Unit Sphere

1. What is the significance of calculating $\int sze^z dS$ on a unit sphere?

Calculating $\int sze^z dS$ on a unit sphere is important in understanding the distribution of a vector field over a spherical surface. It can also be used to find the average value of a function over a unit sphere.

2. How do you set up the integral for $\int sze^z dS$ on a unit sphere?

To set up the integral, you first need to parametrize the unit sphere using spherical coordinates. Then, you can express the function $sze^z$ in terms of these coordinates. Finally, you can use the surface area element $dS$ to set up the integral.

3. What is the general formula for calculating $\int sze^z dS$ on a unit sphere?

The general formula for calculating $\int sze^z dS$ on a unit sphere is:
$\int sze^z dS = \int_{0}^{2\pi}\int_{0}^{\pi} sze^z \sin\phi \, d\phi \, d\theta$
where $s$ is the radius of the sphere, $z$ is the function being integrated, $\phi$ is the colatitude angle, and $\theta$ is the longitude angle.

4. What are some real-world applications of calculating $\int sze^z dS$ on a unit sphere?

One real-world application is in physics, where this calculation is used to determine the flux of a vector field over a spherical surface. It can also be used in atmospheric science to understand the distribution of pollutants or other substances over the Earth's surface.

5. Are there any special techniques or tricks for calculating $\int sze^z dS$ on a unit sphere?

One helpful technique is to use symmetry to simplify the integral. For example, if the function is symmetric with respect to the equator, you can set up the integral for only half of the sphere and then multiply the result by 2. Also, using appropriate substitutions and trigonometric identities can make the integral easier to solve.

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