Surface integrals Definition and 90 Threads

Homework Statement Prove that \int\int_{S} n dS = 0 for any closed surface S.Homework Equations The Attempt at a Solution I used divergence theorem. But i thought it is applicable only if there is another vector multiplied to that outward unit vector (n). \int\int_{S} F {\cdot} n dS

I've seen on most books and in class that Gauss's law is usually stated like \oint \vec{E} \cdot d\vec{A} = \frac{q_{en}}{\epsilon_0} Shouldn't the integral be a surface integral rather than a line integral? I've seen times in problem resolution where they evaluate the integral as a...

Can someone either explain here, or link me to an online document, on how to do surface integrals over surfaces in 3d (not simple ones like planes with x, y, or z held constant). I learned this back in Calculus 2 five years ago, and now I need to do it for my Electrodynamics course and I can't...

Problem: evaluate the double integral of yz over that part of the plane z=y+3 which is inside the cylinder x2+y2=1 I evaluated with respect to z, from z=0 to z=y+3 I got (y3+9y+6y2)/2. Then I integrated this over x2+y2=1. To do that, I switched to polar coordinates, letting x=rcos(theta)...

Homework Statement 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) = xy i + yz j + zx k S is the part of the paraboloid z = 3...

Homework Statement Find the area cut from the surface z = 2xy by the cylinder x^2 + y^2 = 6. [Hint: Set up the integral using rectangular coodinates, then switch to polar coordinates.] Homework Equations A = \iint \sqrt{{z_x}^2+{z_y}^2+1}dxdy = \iint...

How would I calcluate a surface integral in dimensions greater than 3. For example, from the definition of a surfrace integral over a vector field: http://en.wikipedia.org/wiki/Surface_integral#Surface_integrals_of_vector_fields To compute the surface integral, I would first need a vector...

Homework Statement The temperature u in a star of conductivity 6 is inversely proportional to the distance from the center: u = \frac{3}{\sqrt{x^{2} + y^{2} + z^{2}}} If the star is a sphere of radius 3, find the rate of heat flow outward across the surface of the star. Homework...

I'm working on an electrostatics problem that I'd appreciate some clarification on. I'm trying to compute the surface integral of a field \lambda(ix +jy) over a surface that is the half cylinder centered on the origin parallel to the x-axis - that is the end caps of the cylinder are located at...

Homework Statement I am given a vector field associated with a square area of a certain side, let's call this side dx, centered on the x-axis at a certain point, say x = x1. The sides of this cross sectional square are parallel to the axis of y and z. I have to compute the flux of this...

I'm reading Div, Grad Curl, and All That, and in coming up with a formula for the divergence, H.M. Schey starts with a small cube centered at (x,y,z), labels the face parallel to the yz-plane as S1 and calculates \int\int_{S_1}\mathbf{F}\cdot\hat{\mathbf{n}}dS=\int\int_{S_1}F_x(x,y,z)dS...

Homework Statement Find the value of the surface integrals by using the divergence theorem \vec{F} = (y^2z)\vec{i} + (y^3z)\vec{j} + (y^2z^2)\vec{z} S: x^2 + y^2 + z^2 Use spherical coordinates. Homework Equations The Attempt at a Solution I've gotten the integral I think...

Why is it that when we evaluate a surface integral of: f(x, y ,z) over dS, where x = x(u, v) y = y(u, v) z = z(u, v) dS is equal to ||ru X rv|| dA Why don't we use the jacobian here when we change coordinate systems?

Find the surface area of the the portions of the cylinder y^2 + z^2=a^2 bounded by x^2 + y^2 = a^2 not really sure how to go about this. tried to set up a double integral and use polar coordinates but don't know what boundaries to use etc.

Homework Statement F=<0,3,x^2> computer the surface integral over the hemisphere x^2 + y^2 + z^2 = 9 z greater than or equal to 0, outward pointing normal. Homework Equations The Attempt at a Solution I don't know why I keep getting this problem wrong. The general formula for...

Homework Statement calculate the upward flux of f(x,y,z) = <yz,2x+y,y^2+z> Let S be the portion of the cylinder z=4-y^2 lying in the first octant to the right of the plane y=4. a parametrization into the u v plane is:r(u,v)=(u,v,4-v^2) region is a rectangle in the uv plane with bounds, (0,0) ...

You know when a definition is given in terms of z=f(x,y) like the surface integral and its assmed to apply to y=f(x,z)and x=f(y,z) too ... Why is this? I know theyre just variables ...but since x y and z mean something specifically wrt the coordinate system Would it be trivial...

When calculating surface integrals do you have to calculate double integrals for dxdy, dxdz and dydz and add up or what?

Homework Statement Find the surface area of the cone z=3x^2+y^2and above a region in the xy-plane with area 4. Homework Equations double integral sqrt( (dz/dx)^2 + (dz/dy)^2 +1) The Attempt at a Solution I was able to simplify the equation, I just don't know what to do...

[SOLVED] Differential Area Element While doing surface integrals, I am not sure as to which of the following is the correct differential area element to be considered: i] dA = dx dy or ii] A = xy hence, using the product rule: dA = xdy + ydx

I'm in need of help. I need the formulas for total-electric-flux enclosed for a final exam *tommorow. My teacher (nice guy but total slacker) never did any handouts, and I am not the quickest to catch on when he tried to explain in about 20 minutes the concept of surface integrals. I was just...

1) (a past exam question) http://www.geocities.com/asdfasdf23135/advcal30.JPG I am stuck with the parametrizations. For part b, to evaluate the resulting surface integral, I think I need to parametrize the surface. How can I parametrize the surface enclosed by the curve of intersection in...

1) http://www.geocities.com/asdfasdf23135/advcal23.JPG For this question, we have to use the surface integral to compute the area, but I just can't picture what is going on geometrically, so I am stuck at the very begining...can someone please help me? 2)...

I still can't quite see what I'm doing using surface integrals. I'd like an intuitive definition, something like Reinmann sums are to integrals. Believe me, I've seen enough theory with them and I'd rather have a feel for them before I go over the proofs. Here is what I think: You...

Homework Statement Compute the surface integral: g = xyz on x^2+y^2+z^2 = 1 above z^2=x^2+y^2. Homework Equations The Attempt at a Solution I'm only doubtful about the parameterization. Under normal circumstances, since x^2+y^2+z^2 = 1 is a sphere, we can write: r =...

Homework Statement (Q) Find the area of the surface cut from the paraboloid x^2+y+z^2 = 2 by the plane y=0. Homework Equations The Attempt at a Solution The unit normal vector in this case will be j. Moreover, the gradient vector will be sqrt(4x^2+4z^2+1). And the denominator...

Homework Statement (Q) Find the area of the portion of the surface x^2 - 2z = 0 that lies above the triangle bounded by the lines x = sqrt(3), y = 0, and y = x in the xy-plane. Homework Equations The Attempt at a Solution The know how to find the gradient vector. The part...

Homework Statement I have a vector function, and I need to take the surface integral of it over a hemisphere, top half only. I'm "confirming" the divergence theorem by doing a volume integral and surface integral. Already did the volume one so I have something to compare to already. The...

Homework Statement Evaluate ∫∫S √(1 + x^2 + y^2) dS where S is the helicoid: r(u, v) = ucos(v)i + usin(v)j + vk, with 0 ≤ u ≤ 4, 0 ≤ v ≤ 3π The Attempt at a Solution What I tried to do was say x=ucos(v) and y=usin(v), then I plugged those into the sqrt(1+x^2+y^2) eq, which I ended...

Please Help! Surface integrals I am wondering if someone can help me with the following? I am asked to evaluate ∫∫F∙dS where F(x,y,z) = z^2xi + (1/3y^3 +tanz)j + (x^2z+y^2)k and S is the top half of the sphere x^2+y^2+z^2 = 1. ∫∫F∙dS = ∫∫∫divFdV. Here, div F = x^2+y^2+z^2. I know that...

[Can someone help with the following? I am supposed to evaluate the line integral of ∫F∙dr. The curve is oriented counterclockwise as viewed from above. So suppose that F(x,y,z) = (x+y^2)I + (y+z^2)j + (z+x^2)k, and C is the triangle formed by (1,00), (0,1,0), (0,0,1). I know that the...

I am wondering if someone could help me evaluate the following: I am asked to find the surface integral ∫∫ydS where S is part of the paraboloid y = x^2+z^2 that lies inside the cylinder x^2+z^2 = 4. The double integral could be rewritten as ∫∫y*√(4(x^2+z^2)+1)dS, or...

I'm working from H.M. Schey's Div, grad, curl, and all that, and am trying to figure out surface integration. One of the example problems boils down to the following surface integral over a projection, with z = 1 - x - y \sqrt{3} \int \!\!\! \int_R 1 - y \,dx \,dy I made x and y go from 0 to...

Hi eveybody, I'm having trouble in surface integrals. I know already what the double integrals measure; a multivariable function ( drawing surface) over "a region of domain".. Now, the surface integrals are for surfaces given by 2-dependent parametrizations over " the surface". my questions...

Hi, I am studying for finals and I'm having trouble calculating flux over sections of spheres. I can do it using the divergence theorem, but I need to know how to do it without divergence thm also. The problem is, when calculating a vector field such as F(x, y, z) = <z, y, x>, say over...

Given F= (ix+jy) Ln(x^2+y^2) and given S, which is a cylinder of radius r, and height h(in the z axis) evaluate \int\int_s F.n \,ds. It says that you shouldn't need to do any work if you think about it enough. I figured I could find the area of the main part to be 2 \pi r h then multiply that...

Hi, can someone help me out with the following question parts? a) Let W be a compact region in R^3 bounded by a piecewise smooth closed surface S. Let f:W \to R be a C^1 scalar function. Prove that for all constant vectors \mathop c\limits^ \to , \int\limits_{}^{} {\int\limits_S^{} f...

Please help! I'm soo confused with surface integrals and have several to do by tues for my tutorial. I don't really understand how to approach surface integrals! :cry: Could someone give me an over-view and help me through the question below - hopefully then I can manage the rest myself...

At the risk of sounding imbecilic, I'm going to pose this question anyway. If I integral a vector function over a surface {a defined region R on a surface S} then what in fact am I doing? I know it sounds bizarre but I can see the logic of the process to find surface areas..but what does this...

I expected Stokes theorem to make my life easier but these problems are even harder than the normal ones I've been doing. Use Stokes' Theorem to evaluate \int\int_ScurlFdS where F(x, y, z) = < x^2*y^3*z, sin(xyz) ,xyz > S: Part of cone y^2 = x^2 + z^2 that lies between the planes y =...