Surface integral Definition and 253 Threads

Hi. I have this integral \int_0^{2\pi}\int_0^\pi \mathbf A\cdot\hat r d\theta d\phi where \hat r is the position unit vector in spherical coordinates and \mathbf A is a constant vector. Is it possible to evaluate this integral without calculating the dot product explicitly, i.e. without...

Homework Statement I'm looking to do the surface integral of \oint {\vec v \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over a} } where v is arbitrary and in spherical coordinates and the surface is the triangle enclosed by the points (0,0,0) -> (0,1,0) ->...

Homework Statement find the flux of the field F(vector) across the portion of the sphere x^2+y^2+z^2=a^2 in the first octant directed away from the origin Homework Equations F(x,y,z)=zk(hat) The Attempt at a Solution i used Flux=double integral over x-y plane F.n(unit...

I'm self learning and working my way through the book "Div Grad Curl and All That". On one of the pages (27) the author says \int_{ }^{ } \int_{ }^{ } z^2 dS = \int_{ }^{ } \int_{ }^{ } \sqrt[ ]{ 1 - x^2 - y^2 } dx dy "This is an ordinary double integral, and you should verify that its...

Homework Statement Evauate Surface integral \int\int_{\sigma}(x^2 + y^2)dS where \sigma is the portion of the sphere x^2 + y^2 + z^2 = 4 above the plane z = 1. Homework Equations \int\int_{\sigma} f(x,y) \sqrt{\frac{\partial z}{\partial x}^2+\frac{\partial z}{\partial y}^2+1}...

Homework Statement Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 3c^2 within the paraboloid 2cz = x^2 + y^2 using spherical coordinates. (c is a constant)Homework Equations The Attempt at a Solution I converted all the x's to \rho sin\phi cos\theta, y's to \rho sin\phi...

Hi I'm practicing for my exam but I totally suck at the vector fields stuff. I have three questions: 1. Compute the surface integral \int_{}^{} F \cdot dS F vector is=(x,y,z) dS is the area differential Calculate the integral over a hemispherical cap defined by x ^{2}+y ^{2}+z...

use the stokes theorem to evaluate the surface integral [curl F dot dS] where F=(x^2+y^2; x; 2xyz) and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane. I don't see however how to take F dot dr. What is this closed curve dr bounding...

Homework Statement F = (x^2) i + (y^2) j + (z) k, S is the cone z = (x^2+y^2) ^ (1/2), with x^2 +y^2 <= 1, x >= 0, y >= 0, oriented upward. Homework Equations All of the above The Attempt at a Solution My attempted solution is 0. But other students claim that the answers is...

Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + x j + z^2 k S is the helicoid (with upward orientation) with vector...

Homework Statement For the parametrically defined surface S given by r(u,v) = <cos(u+v), sin(u+v), uv>, find the following differential: In double integral over S of f(x, y, z)dS, dS = Homework Equations Above The Attempt at a Solution I thought I needed to put x, y, and...

I want to integrate something in spherical coordinates I have da=R^{2}sin(g)dgdh \hat{r} with g and h angles and \hat{r}=sin(g)cos(h) \hat{i}+sin(g)sin(h) \hat{j}+cos(g) \hat{k} But what is now da_{x}=dydz \hat{i} in spherical coordinates? So I have the expression in ordinary...

Surface Integral Question and Solution Check Hi everyone, this is my first post and I was hoping someone could help me check my solution to this problem (which could be completely wrong) and help me get unstuck at part 3. Any help would be greatly appreciated. Homework Statement...

Homework Statement Prove that the following surface integral for the four slanted faces of a square pyramid with a square base in the xy plane with corners at (0,0) (0,2) (2,0) (2,2) and a top vertex at (1,1,2) is equal to 4 by evaluating it as it stands: \int\int (\nabla \times V)\cdot n dS...

Homework Statement Find the volume integral of the function f=x^{2}+y^{2}+z^{2} over the region inside a sphere of radius R, centered on the origin. Homework Equations Spherical polars x=rsin(\theta)cos(\phi), y=rsin(\theta)sin(\phi), z=rcos(\theta) Jacobian in spherical polars =...

Hi, I'm trying to solve a problem in David Bachman's Geometric Approach to Differential Forms (teaching myself.) The problem is to integrate the scalar function f(x,y,z) = z^2 over the top half of the unit sphere centered at the origin, parameterized by \phi(r,\theta) = (rcos\theta, rsin\theta...

Homework Statement I am trying to sort out surface integrals in my head, and have become more confused when attempting to calculate the surface integral of a hemisphere. I am getting confused about which values to use as boundaries. Homework Equations da=R^2 sinθdθdφ The...

Homework Statement Let S be the part of the paraboloid z=1+x^2+y^2 lying above the rectangle x between 0 and 1; y between -1 and 0 and oriented by the upward normal. Compute \int\int_SF\cdot n\,dS where F=<xz, xy, yz> So I have Parametrized the surface S as r(x,y,z)=<x, y, 1+x2+y2>...

Homework Statement Evaluate the surface Integral I=\int\int_S\vec{F}\cdot\vec{n}\,dS where \vec{F}=<z^2+xy^2,x^100e^x, y+x^2z> and S is the surface bounded by the paraboloid z=x^2+y^2 and the plane z=1; oriented by the outward normal.The Attempt at a Solution...

Homework Statement Evaluate the surface integral \vec{F}\cdot\vec{n}\, dS where \vec{F}=<-y,x,0> and S is the part of the plane z=8x-4y-5 that lies below the triangle with vertices at (0,0,0,), (0,1,0,) and (1,0,0). The orientation of S is given by the upward normal vector. answer: 2...

Homework Statement Find \int\int_{S} F dS where S is determined by z=0, 0\leqx\leq1, 0\leqy\leq1 and F (x,y,z) = xi+x2j-yzk. Homework Equations Tu=\frac{\partial(x)}{\partial(u)}(u,v)i+\frac{\partial(y)}{\partial(u)}(u,v)j+\frac{\partial(z)}{\partial(u)}(u,v)k...

Homework Statement Evaluate: \int\intG(r)dA Where G = z S: x2 + y2 + z2 = 9 z \geq 0 Homework Equations Parameterization x = r sinu cosv y = r sinu sin v z = r cos u The Attempt at a Solution r(u,v) = (r sinu cosv)i + (r...

Homework Statement A log of wood which is approximately circular in cross section has diameter equal to 0.5m. Calculate the maximum area of the rectangular timber section that can be obtained from the log. Homework Equations The Attempt at a Solution No idea of solution

Homework Statement Evaluate the double integral of yz dS. S is the surface with parametric equations x=(u^2), y=usinv, z=ucosv, 0<u<1, 0<v<(pi/2) (all the "less than" signs signify "less than or equal to" here) Homework Equations double integral of dot product of (F) and normal...

Homework Statement This is annoying me because I am so clearly being a muppet somewhere. I need to integrate the vector field (x,-y,z).(vector)ds over the surface of a cyliner x^2 + y^2 < 4 (or equal to) and z is between 0 and 1. The Attempt at a Solution I have to do it both with and...

Homework Statement \int\int{\frac {x}{\sqrt {1+4\,{x}^{2}+4\,{y}^{2}}}}dS Where S is the parabaloid z = 25 - x^{2} -y^{2} that lies within the cylinder x^{2}+(y-1)^{2}=1 The Attempt at a Solution First i use the following: {\it dS}=\sqrt {1+{\frac {{{\it df}}^{2}}{{{\it...

Homework Statement Q] A charge 'Q' is kept over a non-conducting square plate of side 'l' at a height l/2 over the center of the plate. Find the electric flux through the square plate surface. Neglect any induction that may occur. Homework Equations \phi = \int \overrightarrow{E}\cdot...

Homework Statement I'm not sure how to convert this surface integral into a double integral for evaluation. \iint_S \frac{1}{1 + 4(x^2 + y^2)} dS S is the portion of the paraboloid z = x^2 + y^2 between z = 0 and z = 1. The Attempt at a Solution How do you project this onto the xy-plane...

As you know surface integrals are integrated with respect to dS. We then tranform the integral into one in dxdy. Is this the end of the problem or must we calculate it for dxdz and dydz as well and if so do you add up all results at the end!?

May be this should have been in math section but since this came out while studying Electrodynamics i put it here we have \boxed{\int_{S} \nabla \times \vec{B}.d\vec{a}=\oint \vec{B}.d\vec{l}} Q.well there are many areas with the same boundary which one to choose from? well if we know the...

Homework Statement (Q) Compute the surface integral g = xy over the triangle x+y+z=1, x,y,z>=0. Homework Equations The Attempt at a Solution The triangular region basically means that the region in consideration is a plane and not a sphere, cylinder etc... Therefore, we can...

Homework Statement Let F be F = ( x^2 z^2 ) i + (sin xyz) j + (e^x z) k.Find \int\int \nabla \times F \cdot n dS where the region E is above the cone z^2 = x^2 + y^2 and inside the sphere centered at (0,0,1) and with radius 1. (so it is x^2 + y^2 + (z-1)^2 = 1).. I know that they intersect at...

a) Find the area of the part of the surface S = {x^2+ y^2+ (z-1)^2 = 4, 0 ≤ z ≤ 1}. Note that this is part of the sphere of radius 2 with center (0,0,1).

Homework Statement Evaluate [double integral]f.n ds where f=xi+yj-2zk and S is the surface of the sphere x^2+y^2+z^2=a^2 above x-y plane. The Attempt at a Solution I know that the sphere's orthogonal projection has to be taken on the x-y plane,but I'm having trouble with the...

Problem : Evaluate [double integral]f.n ds where f=xi+yj-2zk and S is the surface of the sphere x^2+y^2+z^2=a^2 above x-y plane. My effort:: I know that the sphere's orthogonal projection has to be taken on the x-y plane,but I'm having trouble with the integration.Please help!

i'm trying to understand stoke's theorem and am having trouble seeing whether the surface integral for a given surface changes with any change in its shape, or if it only changes depending on the cross sectional area perpendicular to the direction of the vector field. can anybody help me out?

Okay - I thought that I figured this stuff out, but I didn't. [SIZE="5"]The Problem When G(x, y, z) = (1-x^2-y^2)^{3/2}, and z = \sqrt{1-x^2-y^2}, evaluate the surface integral. [SIZE="5"]My Work I keep trying this but I end up with the following integral that I cannot evaluate...

[SIZE="5"]The Problem Evaluate the surface integral of G(x, y, z) = \frac{1}{1 + 4(x^2+y^2)} where z is the paraboloid defined by z = x^2 + y^2 , from z = 0 to z = 1. [SIZE="5"]My Work I rewrote G(x, y, z) as \frac{1}{1+4z}. Then, I evaluated the surface integral (I'm skipping a few...

Hi, I'm having problems evaluating a surface integral. \int\limits_{}^{} {\int\limits_S^{} {xdS} } where S is the triangle with vertices (1,0,0), (0,2,0) and (0,1,1). I need to parameterise the triangle but I don't know how to. I tried (x,y,z) = (1,0,0) + u[(0,2,0)-(1,0,0)] +...

\int x {(C - x^{2/3})}^{3/2} dx Any ideas?

I am really struggling with this one: Calculate \Int F.ndS , where F = a * x^3 * i + b*y^3*j + c*z^3*k where a,b and c are constants, over the surface of a sphere of radius a, centred at the origin. note that F and n are vectors (sorry, tried to type them in bold...but it...

I have two problems on surface integrals. 1] I have a constant vector \vec v = v_0\hat k. I have to evaluate the flux of this vector field through a curved hemispherical surface defined by x^2 + y^2 + z^2 = r^2, for z>0. The question says use Stoke's theorem. Stoke's theorem suggests...

Hi, can someone give me some assistance with the following questions? 1. Let f(x,y,z), g(x,y,z) and h(x,y,z) be any C^2 scalar functions. Prove that \nabla \bullet \left( {f\nabla g \times \nabla h} \right) = \nabla f \bullet \left( {\nabla g \times \nabla h} \right) . 2. Let S be the...

Everything was going fine until I bumped into this... (b^2*c^2*Cos[x]^2*Sin[y]^4 + a^2*c^2*Sin[y]^4*Sin[x]^2)^(1/2) ...integrate that with respect to y, for the boundaries y=0..Pi. A Jacobian Transformation would be a good start, but I have no idea what functions I would use to simplify...

I need to evaluate the surface integral of f=x over a semi sphere. I know how to evaluate surface integral of a semi sphere but what are my steps in this case. As I found from books I should double integrate f = x with semi sphere limits. The problem is that I don't know how to start and...

Hi! I don't know how to approach this problem. I need a little bit of help please. Here is the problem: Find the surface area of that portion of the sphere x^2 +y^2 + z^2 =a^2 that is above the xy-plane and within the cylinder x^2 + y^2 = b^2, 0 \leq b \leq a

A\; =\; 4\dot{r}\; +\; 3\dot{\theta }\; -\; 2\dot{\phi } Now the surface integral integral is: \int_{}^{}{\left( ?\times A \right)\; •\; da} (the ? mark is a del operator and the dot over a variable means a unit vector) ?\times A\; =\frac{\dot{r}}{r\sin \theta }\left[ \frac{\partial...

"Find the surface integral of r over a surface of a sphere of radius and center at the origin. Also find the volume integral of Gradient•R and compare your results" Do I just integrate r to get (1/2)r^2 and plug some limits in since the r-hats equal one?

Here is the question: Evaluate the surface integral ∫∫s (X^4 + Y^4 + Z^4) dσ, where dσ is the surface element and S = { (X,Y,Z) : X^2 + y^2 + Z^2 = 1} I know you have to take the square root of 1 + (dz/dx)^2 + (dz/dy)^2 dxdy. And I got -2X/2Z and -2Y/2Z, respectively. Then, I must...

Could someone take a look at this please? Thanks ===== Q. Evaluate Integral A.n dS for the following case: A=(6z, 2x+y, -x) and S is the entire surface of the region bounded by the cylinder x^2 + z^2 = 9, x=0, y=0, z=0 and y=8. ===== Using Gauss' (or Divergence) Theorem: Integral A.n...