What is Surface integral: Definition and 260 Discussions
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.
Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.
I am asking this question because my solution does not seem to match the solution at the end of the book (Apostol Vol II, section 12.10, problem 9).
Here is my attempt to solve this problem.
If our coordinate system is chosen such that the z-axis lines up with the axis of the cone then by...
I am trying to compute the stress tensor defined as ##\vec{\Pi}=\eta(\nabla{\vec{u}}+\nabla{\vec{u}}^T)## where ##T## indicates the transpose.
The vector field ##\vec{u}## is defined as follows: ##\vec{u}(\vec{r})=(\frac{a}{r})^3(\vec{\omega} \times \vec{r})## with ##a## being a constant...
I'm supposed to do the surface integral on A by using spherical coordinates.
$$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$
$$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$
Now I'm trying to do
$$\iint A dS = (rsin\theta cos\phi, rsin\theta...
Calculate surface integral ## \displaystyle\iint\limits_S curl F \cdot dS ## where S is the surface, oriented outward in below given figure and F = [ z,2xy,x+y].
How can we answer this question?
Evaluate the surface integral $\iint\limits_{\sum} f \cdot d\sigma $ where $ f(x,y,z) = x^2\hat{i} + xy\hat{j} + z\hat{k}$ and $\sum$ is the part of the plane 6x +3y +2z =6 with x ≥ 0, y ≥ 0,
z ≥ 0 , with the outward unit normal n pointing in the positive z direction.
My attempt to answer...
I start by parametarize the surface with two variables:
$$r(u,v) = (u, v, \frac {d -au -bv} c)$$
The I can get the normal vector by
$$dr/du \times dr/dv$$
What limits should I use to integrate this only within the elipse?
I could redo the whole thing and try write r(u, v) as u being the...
this method of derivation is approximating the function using a polyhedron.
concentrating on one of the surfaces(say the L'th surface which has an area ##\Delta S_l## and let ##(x_l,y_l,z_l)## be the coordinate of the point at which the face is tangent to the surface and let ##\hat n## be the...
Hi,
I am trying to calculate the heat flow across the boundary of a solid cylinder. The cylinder is described by x^2 + y^2 ≤ 1, 1 ≤ z ≤ 4. The temperature at point (x,y,z) in a region containing the cylinder is T(x,y,z) = (x^2 + y^2)z. The thermal conductivity of the cylinder is 55. The...
Hi,
I just had a quick question about a step in the method of calculating the surface integral and why it is valid. I have already done the divergence step and it yields the correct result.
Method:
Let us calculate the normal: ## \nabla (z + x^2 + y^2 - 3) = (2x, 2y, 1) ##. Just to double...
Let ##S_t## be a uniformly expanding hemisphere described by ##x^2+y^2+z^2=(vt)^2, (z\ge0)##
I assume by verify they just want me to calculate this for the surface. I guess that ##\textbf{v}=(x/t,y/t,z/t)## because ##v=\frac{\sqrt{x^2+y^2+z^2}}{t}##. The three terms in the parentheses evaluate...
Note that the solution is 5625 V/m in z direction which is found easier using Gauss' law, but I want to find the same result using Coulombs law for confirmation.
Lets give the radius 0.04 the variable a = 0.04m.
##\rho## is the charge distribution distributed evenly on the surface of the...
Problem Statement: Requesting for re check
Relevant Equations: Requesting for re check
In this eq.A4 putting ##v=Hr+u## the first integrand in eq.A5 is coming as ##H(r(\nabla•u)-(r•\nabla)u+2u)\ne\nabla×(r×u)##
Am I right??
Can I request anyone to please recheck it...
using this the author...
Let ##V'## be the volume of dipole distribution and ##S'## be the boundary.
The potential of a dipole distribution at a point ##P## is:
##\displaystyle\psi=-k \int_{V'}
\dfrac{\vec{\nabla'}.\vec{M'}}{r}dV'
+k \oint_{S'}\dfrac{\vec{M'}.\hat{n}}{r}dS'##
If ##P\in V'## and ##P\in S'##, the...
I want to compute:
$$\oint_{c} F \cdot dr$$
I have done the following:
$$\iint_{R} (\nabla \times v) \cdot n \frac{dxdy}{|n \cdot k|} = \iint (9z-1) dxdy$$
I don't know what limits the surface integral will have. Actually, I am not sure what's the surface.
May you shed some light...
Homework Statement
Find ##\iint_S ydS##, where ##s## is the part of the cone ##z = \sqrt{2(x^2 + y^2)}## that lies below the plane ##z = 1 + y##
Homework EquationsThe Attempt at a Solution
[/B]
I have already posted this question on MSE...
##\mathbf{M'}## is a vector field in volume ##V'## and ##P## be any point on the surface of ##V'## with position vector ##\mathbf {r}##
Now by Gauss divergence theorem:
\begin{align}
\iiint_{V'} \left[ \nabla' . \left( \dfrac{\mathbf{M'}}{\left| \mathbf{r}-\mathbf{r'} \right|}...
Homework Statement
A charge q is placed at one corner of a cube. What is the value of the flux of the charge's electric field through one of its faces?
Homework Equations
The flux surface integral of an electric field is equal to the value of the charge enclosed divided by the epsilon_naught...
Homework Statement
If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##.
The Attempt at a Solution
From gauss divergence theorem we know
##\int...
Homework Statement
∫∫ F ⋅ ndτ over the spherical region x^2 + y^2 + z^2 = 25
given F = r^3 r i already converted the cartesian coordinates to spherical in FHomework Equations
n = r[/B]The Attempt at a Solution
I know I can plug in F into the equation and then dot it with r to get the...
Hi.
If a function f is normalizable ,ie f→0 as | x | → infinity or r→ infinity then I presume the following surface integral f dS over infinite space is zero ?
But I thought about this again and it seems like a case of zero x infinity. The function is zero at the infinite surface but the area...
Homework Statement
Let G=x^2i+xyj+zk And let S be the surface with points connecting (0,0,0) , (1,1,0) and (2,2,2)
Find ∬GdS. (over S)
Homework EquationsThe Attempt at a Solution
I parametrised the surface and found 0=2x-2y. I’m not sure if this is correct. And I’m also uncertain about...
Homework Statement
Calculate
\int_{S} \vec{F} \cdot d\vec{S} where
\vec{F} = z \hat{z} - \frac{x\hat{x} + y \hat{y} }{ x^2 + y^2 }
And S is part of the Ellipsoid x^2 + y^2 + 2z^2 = 4 , z > 0 and the normal directed such that
\vec{n} \cdot \hat{z} > 0
Homework Equations
All the...
From my drawings it seems to be half of hemisphere. Am I right? How can I solve this task?
Determine the flux of the vector field $$ f=(x,(z+y)e^x,-xz^2)^T$$ through the surface $Q(u,w)$, which is defined in the follwoing way:
1) the two boundaries are given by $$\delta...
So the main thing I'm wondering is given a question how do we determine whether to use one of the fundamentals theorems of vector calculus or just directly evaluate the integral, and if usage of one of the theorems is required how do we determine which one to use in the situation? Examples are...
Homework Statement
Solve the surface integral ##\displaystyle \iint_S z^2 \, dS##, where ##S## is the part of the paraboloid ##x=y^2+z^2## given by ##0 \le x \le 1##.
Homework EquationsThe Attempt at a Solution
First, we make the parametrization ##x=u^2+v^2, \, y=u, \, z = v##, so let...
Homework Statement
Let n be the unit outward normal of a spherical surface of Radius R, let the surface of the sphere be denoted by S.
Evalute Surface integral of nndS
Homework EquationsThe Attempt at a Solution
I have evaluated the surface integral of ndS and found it to be 0. but am not...
Let V be the region bounded by the hemisphere z=1-sqrt(1-x^2-y^2) and the plane z=1, and let S be the surface enclosing V. consider the vector field $F= x(z-1)\hat{\imath}+y(z-1)\hat{\jmath}-xy\hat{k}$.
Homework Statement
Is the solution provided by the author wrong ? Stokes theorem is used to calculate the line integral of vector filed , am i right ?
Homework EquationsThe Attempt at a Solution
To find the surface integral of many different planes in a solid , we need to use Gauss theorem ...
I assume this is a simple summation of the normal components of the vector fields at the given points multiplied by dA which in this case would be 1/4.
This is not being accepted as the correct answer. Not sure where I am going wrong. My textbook doesn't discuss estimating surface integrals...
Homework Statement
I'm just required to setup the integral for the question posted below
Homework EquationsThe Attempt at a Solution
So solving for phi @ the intersection of the sphere and the plane z=2:
z = pcos(phi)
2 = 3cos(phi)
phi = arccos(2/3)
so my limits for phi would go from 0 to...
Homework Statement
Let S be the portion of the paraboloid ##z = 4 - x^2 - y^2 ## that lies above the plane ##z = 0## and let ##\vec F = < z-y, x+z, -e^{ xyz }cos y >##. Use Stoke's Theorem to find the surface integral ##\iint_S (\nabla × \vec F) ⋅ \vec n \,dS##.
Homework Equations
##\iint_S...
Homework Statement
Hi everybody! I'm currently training at surface integrals of vector fields, and I'd like to check if my results are correct AND if there is any shortcut possible in the method I use. I'm preparing for an exam, and I found that it takes me way too much time to solve it. I...
Homework Statement
It is evaluating a surface integral.
Homework Equations
∫s∫ f(x,y,z) dS = ∫R∫ f[x,y,g(x,y)]√(1+[gx(x,y)]2+[gy(x,y)]2) dA
The Attempt at a Solution
I set z=g(x) and found my partial derivatives to be gx=√x, and gy=0. I then inserted them back into the radical and came up...
Homework Statement
when the normal vector n is oriented upward , why the dz/dx and dz/dy is negative ? shouldn't the k = positive , while the dz/dx and dz/dy is also positive?
Homework EquationsThe Attempt at a Solution
is the author wrong ? [/B]
I am a tenth grader, and a newbie to Advanced Calculus. While working out problems sets for Gauss's Law, I encountered the following Surface Integral:
I couldn't attempt anything, having no knowledge over surface integration. So please help.
Homework Statement
Problem is in image uploaded
Homework Equations
n/a
The Attempt at a Solution
x = u, y = v and z = 1 - u - v
∂r/∂u × ∂r/∂v = i + j + k
F dot N = u^2 + 3v^2
∫∫(u^2 + 3v^2 )dudv
My problem is I'm not sure what I should take as the limits?
Should I flip around the order of...
Homework Statement
I have this problem in an online assignment. Someone told me the answer, so I already got it right, but I don't know why my logic leads me to the wrong answer. The problem:
The temperature u of a star of conductivity 1 is defined by u = \frac{1}{sqrt(x^2+y^2+z^2)}. If the...
Homework Statement
Evaluate integral A.n dS for A=(y,2x,-z) and S is the surface of the plane 2x+y=6 in the first octant of the plane cut off by z=4
Homework Equations
Integral A.n dS
The Attempt at a Solution
The normal to the plane is (2,1,0) so the unit normal vector is 1/sqrt3 (2,1,0)...
Hey! :o
I want to calculate the surface integral of $$F(x,y,z)=(0,0,z)$$ on the unit sphere with parametrization
$$x=\sin u \cos v, \ y=\sin u \sin v , \ z=\cos u \\ 0\leq u\leq \pi, \ 0\leq v\leq 2\pi$$
with positive direction the direction of $T_u\times T_v$. Could you give some hints how...
Homework Statement
Find the area of the part of z^2=xy that lies inside the hemisphere x^2+y^2+z^2=1, z>0
Homework Equations
da= double integral sqrt(1+(dz/dx)^2+(dz/dy)^2))dxdy
The Attempt at a Solution
(dz/dx)^2=y/2x
(dz/dy)^2=x/2y
=> double integral (x+y)(sqrt(2xy)^-1/5) dxdy
Now I'm...
Joos asserts on page 31 https://books.google.com/books?id=btrCAgAAQBAJ&lpg=PP1&pg=PA31#v=onepage&q&f=false that
$$\nabla \times \mathfrak{v} = \lim_{\Delta \tau \to 0} \frac{1}{\Delta \tau }\oint d\mathfrak{S}\times \mathfrak{v}$$
I tried to demonstrate this, and neglected to place the surface...
The following is my interpretation of the development of the divergence of a vector field given by Joos:
$$dy dz dv_x=dy dz\left(v_x(dx)-v_x(0)\right)=dy dz\left(v_x(0)+dx\frac{\partial v_x}{\partial x}(0)- v_x(0)\right)$$
$$=dy dz dx\frac{\partial v_x}{\partial x}(0)=d\tau \frac{\partial...
Homework Statement
Calculate the integral ##\oint_C \vec F \cdot d\vec S##, where ##C## is the closed curve constructed by the intersection of the surfaces ##z = \frac{x^2+y^2}{4a}## and ##x^2+y^2+z^2=9a^2##, and ##\vec F## is the field ##\vec F = F_0\left( \frac{a}{\rho}+\frac{\rho^2}{a^2}...
Homework Statement
Homework Equations
The path integral equation, Stokes Theorem, the curl
The Attempt at a Solution
[/B]
sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First...
Hi everyone,
I need some help to look if I did these calculations right.Let us assume a three dimensional magnetic field:
##\vec{B}(x,y,z) = B_x(x,y,z)\hat{x} + B_y(x,y,z)\hat{y} + B_z(x,y,z)\hat{z}##
The equation for the force on a superconducting particle in a magnetic field is given by...
I'm beyond multi-variable calculus, where this is taught, but I still don't know what the hell a surface integral is. I understand that d\sigma is the surface element, and | \frac{\partial \vec{r}}{\partial u}du \times \frac{\partial \vec{r}}{\partial v}dv | = d\sigma = |\frac{\partial...