# Questions tagged [surface-integrals]

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral.

1,281 questions
Filter by
Sorted by
Tagged with
52 views

17 views

• 507
30 views

42 views

### surface of $M=\{(x,y,z)\in\mathbb R^3: x^2+y^2<1 \text{ and } z=3-2x-2y\}$

For $M=\{(x,y,z)\in\mathbb R^3: x^2+y^2<1 \text{ and } z=3-2x-2y\}$ I want to determine the surface of $M$, this means I want to calculate $\int_M 1 dS_M$. I don't know how to this and I only need ...
• 284
42 views

• 13
52 views

### How do you calculate the fluid flowing through a pipe using a flux integral?

I am beginning to learn vector calculus and over the last few days I have been looking at integrals. An example the lecturer used was fluid flow. He said that the volume of fluid flowing through a ...
1 vote
12 views

### Estimate flux integral of $\frac{1}{|x|}$ in $\Bbb{R}^3$ through a quadratic surface.

My question occured reading this paper in the first example of section 6. Let $f:\Bbb{R}^3 \rightarrow \Bbb{R}^3$, $f(x)=|x|$ a vector field. Now divide $\Bbb{R}^3$ up into disjoint unit cubes ...
38 views

### Surface integral ( limits of surface)

Calculate $$\iint_{S} \frac{xy}{\sqrt{1+2x^{2}}} dS$$ where $$S =\{(x, y, x^{2}+y ) , 0 \leq x \leq y, x+y \leq 1\}$$ My attempt is this i cannot figure out the limits of $x$ and $y$ In the ans it ...
• 21
24 views

### Surface integral of a sphere via definition

I attempted to calculate the surface integral $\iint_S F \cdot n \ dS$ where $F = (x^2 + y^2 + z^2)(x, y, z)$ and $S$ is the sphere of radius $a$ centered at the origin. I used the divergence theorem ...
18 views

### computing the flux integral on this surface

Compute the flux integral for the field $\mathbf{F} = x\mathbf{i} + y\mathbf{j} + 5\mathbf{k}$, where $S$ is the boundary of the cylinder enclosed by $x^2+z^2=1$ and the planes $y=0$ and $x+y=2$. How ...
• 1,723
1 vote
40 views

### Flux of $\mathbf{F}=(xz,yz,x^2+y^2)$ across paraboloid $z=4-x^2-y^2,\ z\geq 0$

I have calculated the flux of the vector field $\mathbf{F}=\begin{bmatrix}xz\\ yz\\ x^2+y^2\end{bmatrix}$ outward across the surface of the paraboloid $S$ given by $z=4-x^2-y^2,\ z\geq 0$ (with ...
• 3,330
39 views

1 vote
29 views

### A Question about evaluating surface integrals

Vector calculus is relatively new to me so I have a little trouble understanding double integrals intuitively. The question is esssentially this: Calculate the surface integral of $\vec v=x^2 \hat j$...
25 views

### Prove $\frac{1}{v(B_{\varepsilon})}\iint_{S_{\varepsilon}}zf{\rm d}S\underset{\varepsilon\rightarrow0}{\rightarrow}\frac{\partial f}{\partial z}$

Let $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a continuously differentiable function. Let $S_\varepsilon,B_\varepsilon$ be the sphere and ball of radius $\varepsilon$ around the origin, respectively. ...
• 3,247
34 views

91 views

### Divergence Theorem over a Simple Smooth Surface and at its Boundary

I have to solve the following theoretical problem: In $\mathbb{R}^3$ with a fixed Cartesian coordinate system $(x,y,z)$, let $D$ be an open subset in the $(x,y)$ plane. $D$ can be considered as a ...
• 115
43 views

I need to transform the following integral into a surface integral (if that's possible): $$\int\int\int_\Omega R\times (\nabla \times A) dv = \int\int_{\partial \Omega} ? . {\bf n} da,$$ where $R = (... • 1,701 0 votes 0 answers 47 views ### Surface integral of vector field$\vec{F}=\langle 2x^3,5xz^2,x^2+6y^2z \rangle$I want to evaluate the surface intgeral of $$\vec{F}=\langle 2x^3,5xz^2,x^2+6y^2z\rangle$$ in the$xy$-plane and$x^2+y^2\le4$. I know I should use$$\iint_S\vec F\cdot d\vec{S}=\iint_S\vec F\cdot \... 2 votes 0 answers 62 views ### Surface Integral of Vector Field$\textbf{A}=(xz,0,yz)$I have to solve the following problem: A cylinder of radius R in$\mathbb{R}^3$with axis along the$z$-axis of a Cartesian coordinate system$(x,y,z)$can be parametrised as$\textbf{x}(\theta,z) = (...
• 115
I'm studying Do Carmo's differential geometry. Let $Q$ be a compact region in $\mathbb{R^2}$ that is contained in a coordinate nbd $X : U \rightarrow S$, where $S$ is regular surface. I want to get ...
Computing the area of, or integrating over, a rectangle aligned with the x and y axes is really straightforward. Indeed, if $x \in [-\frac{b}{2},\frac{b}{2}]$ and $y \in [-\frac{y}{2},\frac{y}{2}]$, ...