Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

Read more on wikipedia's entry Surface integral.

1384 questions
2
votes
2 answers

Need help solving a surface integral question

The question is: I understand the method used in the book. However, when I try a slightly different method, I end up with an incorrect answer: 2$\pi$ instead of 4$\pi$ as given in the book. I have already tried looking for what I could be going…
MathematiciansWalkIntoABar
  • 23
1
vote
1 answer

How to prove this formula for surface integrals?

Let $f(x)$ be a continuous function on $\mathbb{R}$. Prove $$\iint\limits_{S}f(x+y+z)dS=2\pi\int_{-1}^{1}f(\sqrt{3}\xi) d\xi$$ where S is $x^2+y^2+z^2=1$. I have already noticed that $\iint\limits_{S}f(x+y+z)dS=\iint\limits_{S}f(x+y+z)\cdot…
ZTR05
  • 13
1
vote
0 answers

Solving surface integral with projection method

Given surface integral $$ \iint_{S(V)} y^2dzdx + z^2dxdy $$ Where $ S(V) $ is $$ x^2 + y^2 + z^2 = 4 $$ $$ x \geq 0,\quad y \geq 0, \quad z \geq 0$$ If I haven't made any mistake, using Gauss–Ostrogradsky formula outer surface area should be $ 4\pi…
taseo
  • 11
1
vote
1 answer

Does anyone know what kind of surface is defined by $z=\left(x^2+y^2+z^2\right)^2$?

Does anyone know what kind of surface is defined by this equation? $$z=\left(x^2+y^2+z^2\right)^2$$ Thank you in advance.
fourier
  • 93
1
vote
1 answer

Surface integral first kind

I have to calculate surface integral:$$\iint_D zdD$$ where $S$ is part of cone $z=\sqrt{x^2+y^2}, 1
Trevor
  • 533
1
vote
1 answer

Calculating surface integral with sphere

I need to calculate the following surface integral:$\iint x^2dydz+y^2dxdz+z^2dxdy$ on the region $S:(x-a)^2+(y-b)^2+(z-c)^2=R^2$ in the direction of outside normal. So, first i found that normal has coordinates…
Trevor
  • 533
1
vote
0 answers

Mistake in calculating a surface integral

I am trying to solve the following question, which is a standard Surface Integral/Stokes' theorem question. Unfortunately, I've tried to calculate it several times, and each time get the answer slightly wrong. I calculated $\nabla \times B =…
riemann_lebesgue
  • 626
1
vote
1 answer

Calculate a line integral

I need to calculate a line integral: $\int_\gamma (2x+1)dx+(3xy+1)dy$, where $\gamma$ is the surface of the traingle $OAB$ where $O(0,0)\ \ \ A(3,-3)\ \ \ B(-6,-6) $. I tried to solve this by parametrizing the lines $AB$, $AO$ and $OB$, but it turns…
Raducu Mihai
  • 803
1
vote
0 answers

What does a negative surface integral mean?

My try: The surface integral could be interpreted as the flux of a fluid, i.e., the amount of fluid flowing through the surface per unit time. So does a negative or positive sign actually mean the direction of the flow? Or it has something else to…
Manjoy Das
  • 1,006
  • 7
  • 21
1
vote
0 answers

Symmetry in surface integrals on spheres

I stumbled upon an example on surface integrals that uses symmetry, and I feel like I'm missing something that should be simple. We're dealing with a sphere $\Sigma_R$ centered on the origin with radius R. Halfway through the example, we're met with…
DealerUmbra
  • 41
1
vote
1 answer

Solving the surface Integral Using Gauss Divergence Theorem.

Evaluate $\int Fnds$ over the entire surface of the region bounded above $xy$ plane bounded by the cone $z^2 = x^2 + y^2$ and the plane $z =4$ If $F = \hat{i} +\hat{j} - 3\hat{k}$ then Find $\int Fnds$ My question Is can I use Gauss Divergence…
user435638
1
vote
1 answer

Surface Area of $S =$ {$x^2+y^2+z^2=4, (x-1)^2+y^2\leq1$}

Find the area of the following surface, $S =$ {$x^2+y^2+z^2=4, (x-1)^2+y^2\leq1$}? How do I find a parameterization of the surface?
laser01
  • 1,180
1
vote
1 answer

Fundamental vector product definition

In class, we defined the fundamental vector product of $r$, where $$r(u,v) = (X(u,v), Y(u,v), Z(u,v)),$$ as shown in the image. I understand how we got everything in the first line, but how are we going from the three determinants at the end of the…
Euclid
  • 45
  • 7
1
vote
1 answer

Earth's surface area

Here we are trying to calculate the earth's surface area via geodetic coordinates: $x=(Rp(\lambda)+h)\sin (\lambda)\cos(\phi)$ $y=(Rp(\lambda)+h)\sin (\lambda)\sin(\phi)$ $x=((1-e^2)Rp(\lambda)+h)\cos (\lambda)$ where $p(\lambda)=\frac 1…
Leyla Alkan
  • 2,451
1
vote
1 answer

Surface integral for area calculation

Is my procedure correct? Calculate paraboloid area portion of equations $$P \equiv (u \cos v, u \sin v, u^2) $$ with $0 \leq v \leq \dfrac{\pi}{4}$ and $0 \leq u \leq \dfrac{1}{2}\tan v$ This is my attempt: $\bullet$ vector first…
user3204810
  • 299
1
2
3 4 5 6 7