Surface integral Definition and 262 Threads

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...

Hello! This is a question from one of our past exams, and it's had me stumped for the past hour. The question states: The cylinder x^2+y^2=2x cuts out a portion of a surface S from the upper nappe of the cone x^2+y^2=z^2. Compute the surface integral: \int\int (x^4-y^4+y^2z^2-z^2x^2+1)...

Curl and Surface Integral (help!) Hello people! I've been working on this problem, but I can't find how differentials of V on the left side of the equation appear. *** Show, by expansion of the surface integral, that (see attached image). Hint: choose the volume to be a differential...

Yo guys, I'm stumped on how to parameterize this surface and then compute an integral over it I'm supposed to compute \int\int_S \vec{r} over the surface formed by the x-y plane and z=4-(x^2+y^2), but I don't know to put it together and do it (No matter how you work with x and y, z will...