# Tagged Questions

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### Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
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### Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
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### Relationship between Surface Area and Volume

Question: Is there a general relationship between surface area and volume analogous to the below examples? Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ...
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### Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
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### What is the meaning of $d\vec S$ in a surface integral?

Can someone explain if I have a surface $z= 9-x^2-y^2$ What would $\vec{n}$ be? What would $d\vec{S}$ be? Why is $d\vec{S}$ $(2x,2y,1)$ and not $(2x,2y,1)/\sqrt{4x^2+4y^2+1}$? Thanks!
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### Area of the portion of the cylinder $x^2+y^2 = 9$ for which $-1 \leq z \leq 2$ and $0 \leq \theta \leq \pi/2$

Problem: Find the area of the portion of the cylinder $x^2+y^2 = 9$, for which $-1 \leq z \leq 2$ and $0 \leq \theta \leq \pi/2$ I first solved this by parametrizing the surface. $x = 3\cos(u)$ , ...
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### Verification of the Stokes theorem for the surface that is a part of a cone

Let $S$ consist of the part of the cone $z=(x^2+y^2)^{1/2}$ for $x^2+y^2\leq9$ and suppose $${\bf A}=(-y,x,-xyz).$$ Verify that Stokes theorem is satisfied for this choice of $\bf A$ and $S$. In ...
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### Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
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### Understanding surface integrals

My question is a bit vague, but I'm trying to get a better understanding of surface integrals and their relation to physics. Suppose I have a surface, say a sphere, and I have a function which gives ...
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### Surface: intersection of 2 polar curves

I have these two polar curves: $$C_1: r = 2 - \cos(\theta)\\ C_2: r = 3 \cos(\theta)$$ Plots: C1 and C2. I need to find the surface of $D = C_1 \cap C_2$. I started by finding the solution to ...
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### Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
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### Surface Area Line integral problem

I'm trying to figure out how to solve a surface area with surface and line integrals (showing both methods). The area I'm trying to compute is the area of the shape $$x^2+y^2=9$$ bounded by $z=0$ and ...
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### Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
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### Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...