Vector calculus Definition and 422 Threads

Homework Statement The capacity C of an object is the integral over its surface -\int_S \frac{\partial \phi}{\partial n} dA, where the potential φ(x) satisfies Laplace’s equation in the volume outside the object, \phi = 1 on S and \phi \to 0 at \infty . Show that the capacity of a...

I've written all the questions in the PDF file ... Please help me !

okay i was reading the book "div, grad cur and all that" i've just started for my exam. but i got stuck in the beginning. I'm attaching the page because i really don't get it . basically pages 14 and 15. i don't get how they calculate the components of u and v. if someone explains u i'll...

has anyone here studied vector calculus?

I am having problems in finding the limits in surface integral.. for example in case of sphere what will be the limits of theta and phi. somebody please answer quickly. Thanks

Homework Statement show that grad(r.\hat{k}/r^3) = [r^2\hat{k}-3(r.\hat{k})r]/r^5 Homework Equations The Attempt at a Solution I know that r=xi+yj+zk and i know how to calculate the grad from the formula but, what is r.\hat{k} ? thank you

For divergance thereom, say i have a volume integral to calculate of form \iiint_V \nabla F dV i can relate it to the form: \iint_S F.dS = \iiint_V \nabla F dV and calculate using the left hand side, \iint_S F.n dS where n is the unit vector normal to the surface of...

Homework Statement Let S be the surface given by the graph z = 4 - x2 - y2 above the xy-plane (that it is, where z \geq 0) with downward pointing normal, and let F (x,y,z) = xcosz i - ycosz j + (x2 + y2 ) k Compute \oint\oints\oints F dS. (F has a downward pointing normal) (Hint...

Urgent help! Vector Calculus question... Let S be the surface given by the graph z = 4 - x2 - y2 above the xy-plane (that it is, where z \geq 0) with downward pointing normal, and let F (x,y,z) = xcosz i - ycosz j + (x2 + y2 ) k Compute \oint\ointsF dS. (F has a downward pointing normal)...

Hey i was wondering about a paticular problem i found in a textbook. Specifically just one little niggle i have with it. i am given that the electrostatic potential energy in a region of space is given by v(r) = ((q*n)/(epsilon-0))(x^2+y^2) (where n is a constant of dimensions m^-3)...

Homework Statement Suppose we have the vector field F whose x component is given by F_{x}=Ax and whose divergence is known to be zero \vec{\nabla}\cdot\vec{F}=0, then find a possible y component for this field. How many y components are possible? 2. The attempt at a solution So the...

Equation for plane that passes through the point (2,-1,3) and is perpendicular to the linev=(1,-2,2)+t(3,-2,4)? I'm not exactly sure where I'm supposed to go since the only examples in my book show the plane perpendicular to a vector written in a different form. Not in this one

I know this has been posted before, and I've read the post concerning the same problem and I've googled this a million times, but i can't seem to get it. So here's the problem: Given a Point and a Line, find the Line that passes through the point (3,1,-2) and is perpendicular to the line...

Homework Statement I was studying for my finals, and then the only thing that I got stuck on was parameterizing a surface and finding the area of the surface. and my problem states that x2 + y2 + z2 = 4 and z \geq\sqrt{2} Homework Equations so when parameterizing a sphere, it comes...

G'day, If you're given a vector q and have that del x p=q (i.e curl(p)=q), how would you find p? Also for divergences. cheers

Need help checking if my reasoning is sound for this. Homework Statement Isobars are lines of constant temperature. Show that isobars are perpendicular to any part of the boundary that is insulated. Homework Equations u(t,\underline{X}) is the temperature at time t and spatial...

i think i get this now but just checking as i have the exam tomorrow and won't do this part if I am getting it wrong. (2) (a) find a vector perpendicular to a=i+2j-2k and b=-2i+3j+5k just use the cross product? and get 15i+j+7k (b) (i) find the equation of the plane through position...

two questions that i can't ever remember covering first of all finding a perpendicular vector find a vector perpendicular to the vectors a=i+2j-2k and b=-2i+3j+5k and secondly the equation of a plane?? through point with position vector (2,1,1) and perpendicular to (3,-1,2) what are the...

1) Let S be the first-octant portion of the paraboloid z = x^2 + y^2 that is cut off by the plane z=4. If F(x,y,z) = (x^2 + z)i + (y^2z)j + (x^2 + y^2 + z)k , find the flux of F through S. 2) Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y =...

Homework Statement A region R is bounded by the curves y = 12.x and y=5.x^2 If I = (5/12).x^2 .y i + (y/12.x) j find the contribution to the line integral Integral I.dl = Integral (I(x) dx + I(y) dy) taken in the anti clockwise direction with respect to the region R along the curve...

There is a section that contains dealings with Maxwell's equations in my vector calc book, and there are, to my belief, numerous errors. Some of them I am certain on, but the is one I am not so sure on, although it would be a bigger blunder. The book describes Maxwell's equations as, having E...

I don't even know where to start this one. I can do all the other problems in the section, but this one makes no sense

If you start with the two dimensional green's theorem, and you want to extend this three dimensions. F=<P,Q> Closed line integral = Surface Integral of the partials (dP/dx + dQ/dy) da seems to leads the divergence theorem, When the space is extended to three dimensions. On the...

If you start with the two dimensional green's theorem, and you want to extend this three dimensions. F=<P,Q> Closed line integral = Surface Integral of the partials (dP/dx + dQ/dy) da seems to leads the divergence theorem, When the space is extended to three dimensions. On the...

1. (a) Write a formula for the number in terms of the perimeter L and the area A of a circle. (b) Write the differential for your answer in part (a). (c) Suppose that L and A are determined experimentally. Write the resulting relative error in using your answer in part (b). 3...

I'm having a hard time proving that if A=k \frac{m\times r}{(r\cdot r)^{3/2}} (the vectorpotential of a magnetic dipole with moment m), then: B=\nabla\times A=k\frac{3e_r(e_r\cdot m) -m}{(r\cdot r)^{3/2}} without writing the whole thing in components, which becomes long, messy and ugly...

Compute the line integral \int_{C} F\cdot dr where F = -y i + x j. The directed path C in the xy-plane consists of two parts: i) a left semicircle from (0, -1) to (0, 1) with center at the origin, and ii) a straight line segment from (0,1) to (2,1). i) r(t) = cos t i + sin t j [pi/2 <=t<=...

Suppose that F is an inverse square force field; this is, F(r) = cr/ |r|^{3} for some constant c, where r = xi + yj + zmbfk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from these points to the origin. Not exactly...

Find the volume V of the solid under the surface z=4-x^2-y^2 and over the rectangle R consisting of all points (x,y) such that 0<=x<=1 and 0<=y<=2. I have started, but am unsure if my approach is correct or not. x = 4-x^2-y^2 \int^{2}_{0}\int^{1}_{0} 4-x^{2}-y^{2} dx dy is this...

Find the volume V of the solid under the surface z=4-x^2-y^2 and over the rectangle R consisting of all points (x,y) such that 0<=x<=1 and 0<=y<=2. I have started, but am unsure if my approach is correct or not. x = 4-x^2-y^2 \int^{2}_{0}\int^{1}_{0} 4-x^{2}-y^{2} dx dy is this correct?

Find the area inside the lemniscate ,which is described by the equation (x^2 + y^2) = 2a^2 (x^2 - y^2) I have no idea where to start so I found out what a lemniscate is.. but this didn't help me much. I have rearranged the equation for a: a = sqrt ( (x^2 + y^2)^2 / 2(x^2 - y^2) ) a = x^2...

hey guys & gals ... i need your advice in selecting the best possible Vector Calculus book for a physics major planning on pursuing his Master's and perhaps more in the future ... i have been away from math and physics for some time (~ 4 years) and do not have a solid Vec-Calc background...

Hi there is there a tutorial or post explaining vector calculus subscript notation please? e.g. Eijk Kklm dil djm etc etc is there a tutorial explaining these thoroughly and how these can convert into div grad and curl?? i've used the search engine but can't seem to find them. thnx

I'm having trouble with electromagnetic waves, perhaps just a vector calculus issue. I'd much appreciate some help in idenfiying it. If given say an example in an assignment of an electromagnetic wave: E = E_0 cos (omega(sqrt(sigma.mu) z - t )) X + E_0 sin...

what does \left (\vec{A}\cdot \vec\nabla \right ) \vec B mean?

Hi everyone I'm just about to begin a course in multivariate and vector calculus what prior knowledge in maths is good to go over to help me along the way in this course?

Hey guys, it's my first post here so please don't chew my head off if I do something forbidden, hahah. Homework Statement Prove the Law of Sines using Vector Methods. Homework Equations sin(A)/a = sin(B)/b = sin(C)/c The Attempt at a Solution Since axb=sin(C), I decided to...

Homework Statement Find \int_{s} \vec{A} \cdot d\vec{a} given \vec{A} = ( x\hat{i} + y\hat{j} + z\hat{k} ) ( x^2 + y^2 + z^2 ) and the surface S is defined by the sphere R^2 = x^2 + y^2 + z^2 directly and by Gauss's theorem. Homework Equations \int_{s} \vec{A} \cdot d\vec{a} =...

Homework Statement At time t = 0, the vectors \textbf{E} and \textbf{B} are given by \textbf{E} = \textbf{E}_0 and \textbf{B} = \textbf{B}_0 , where te unit vectors, \textbf{E}_0 and \textbf{B}_0 are fixed and orthogonal. The equations of motion are...

I have a problem regarding the function f (x,y) = {x*y*(x^2-y^2)/(x^2+y^2) if (x,y)!=(0,0) and f(x,y)=0 if (x,y)=(0,0). I am asked if this function is differentiable. Running it through a graphing program it looks differentiable. I know the partial derivatives of it in terms of x and y are...

This isn't directly a request for homework help, since classes won't be starting for another two months, but I suppose it will be helpful to homework because I'll be taking Applied Analysis, Mechanics, and Electromagnetism, all of which include vector calculus. Given time, I will try to work...

Dear all I'm a newbie here...having some prob and hope to have your expert advice on how to solve the equation attached. Thx in advance!

Does anyone know how do #1? I thought I just needed to find the binormal and it will give me the equation of the tangent plane, however the calculations are too insane for that to be the way to solve this. I can't think of any other way, since we're dealing with a vector function and not the...

Hi, can someone provide some suggestions? I'm stuck on the following questions. Q. In this question we will consider the consequences if photons had mass. For massive photons the Laplace equation for the electric potential is replaced by \nabla ^2 \Phi = m^2 \Phi . (*) a) Use spherical...

I want vector calculus formulae tables, such as \mathbi{a}\times(\mathbi{b}\times\mathbi{c}) = \mathbi{b}(\mathbi{a} \cdot \mathbi{c} ) - \mathbi{c} (\mathbi{a}\cdot \mathbi{b}) and \nabla \cdot (\mathbi{a} \times \mathbi{b}) = \mathbi{b} \cdot (\nabla \times \mathbi{a}) - \mathbi{a} \cdot...

just help me to check homework on vector calculus :) Find the equation of the normal vector, tangent plane, and the normal line to the graph of the equation at point P. here's the equation and my solution: http://www.mrnerdy.com/forum_img/p1.JPG...

I'm not sure what the question is asking. Any help on getting started with this would be cool. Q: Show that u = r\cos \psi +\frac{1}{2}r^{-2}\cos \psi satisfies \nabla^2 u and also \frac{\partial u}{\partial r}=0 on the unit sphere. Find the velocity field \vec v = \nabla u for flow...

I have a point in (x,y,z) and a plane in the form Ax+By+Cz=D and I need to find the distance between the point and the plane. I tried using parametrics and cross products, but I couldn't figure it out. Thanks

The problem reads as follows: "The projection of a point P = (x,y,z) to a plane is a point on the plane that is closest to P. If the plane is defined by a point P0 = (x0,y0,z0) and a normal vector n=(x1,y1,z1), computer the projection of P on this plane." Well, I haven't had a relevant...

Hi. I have this exam in vector calculus tomorrow, but I'm having trouble sorting the following formula out. Could someone help me on the track or show me why this is an equality. Feels meaningsless to merely memorize the formula. \nabla \times (\bar{u} \times \bar{v}) = (\bar{v} \cdot \nabla)...