Vector calculus Definition and 416 Threads

Hi guys, recently I am reading Kleppner's Mechanics about center of mass, well, I am always a fairly fast learner, but I really got stick here. you see, in the page 119 example 3.3 (in short, a rod with nonuniform density ) let Q(x) be the function density of location vector it said...

Homework Statement Consider a long straight river flowing north with parallel banks 40m apart. Let us use the function u(x) = 3 sin(\pix/40) to model the rate of water flow x units from the west bank.A boat proceeds at a constant speed of 5m/s from a point A on the west bank while maintaining...

Homework Statement An object moves according to this : r(t)= (1+t^2)i + (1-t)j + (t+t^3)k How fast does it moves far from O(0,0) for t=1? Is this velocity the same with the object's velocity? The Attempt at a Solution I can't understand the second part

Homework Statement A particle is constrained to move around the unit circle in the xy plane according to the formula (x,y,z)=(cos(t2),sin(t2),0), t\geq0. At what point on the circle should the particle be released to hit a target at (2,0,0)? Homework Equations None The...

Homework Statement A force is applied to a particle, defined by: F(x,y)= (y^2, 2xy) << This is a verticle bracket with the y^2 ontop of the 2xy The path of the particle is straight. The particle moves from (-1,2) to (1,3) i) Calculate the work that the force F does as the...

Homework Statement 1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2) A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards. B)Verify Gauss's divergence theorem...

Homework Statement A curve C in space is defined implicitly on the cylinder x^2+y^2=1 by the additional equation: x^2-xy+y^2-z^2=1. Find the point or points on C closest to the origin.Homework Equations d = ((x-x0)+(y-y0)+(z-z0))^(1/2) - This is the distance formula. Please note that I did...

Hello, I am a physics major currently finished with my second year and I am trying to teach myself vector calculus since the Calculus III class at my university did not include it and I am taking upper level electromagnetic physics courses at another univeristy this coming up fall semester. So...

Homework Statement A smooth vector field F has divF(1,2,3) = 5. Estimate the flux of F out of a small sphere of radius 0.01 centered at the point (1,2,3). Homework Equations Cartesian Coordinate Definition of Divergence: If F= F1i + F2j +F3k, then divF=dF1/dx + dF2/dy + dF3/dz The...

Homework Statement F = 2i + 3j through a disk of radius 5 in the plane y = 2 oriented in the direction of increasing y. Calculate the flux of the vector field through the surface. Homework Equations The Attempt at a Solution I know that I need to calculate the area vector...

Homework Statement Can anyone help me solve couple(if not all of these problems on this practice test? I have this huge test thursday and I can't seem to get any of these concepts thru my head. You might not be able to solve the first set of problems(1-6) because of a different teachers...

Homework Statement Show that v\nablav = \nablaxvxv v · ∇v = ∇(0.5v2 + c × v c=∇ × v My attempt ∇(A · B)= B · ∇A + A · ∇B + B×(∇×A) + A×(∇×B) Replace A and B with V ∇(v · v)= v · ∇v + v · ∇v + v×(∇×v) + v×(∇×v) v · ∇v = ∇(0.5v2 - v×(∇×v) Is v×(∇×v) = =∇ × v × v? And...

Homework Statement Two forces act on an object at an angle of 50°. One force is 150 N. The resultant force is 200 N. Find the second force and the angle that it makes with the resultant, using only cartesian vectors. Homework Equations The Attempt at a Solution Over here, I am very confused...

Homework Statement The displacements of two ships, A and B, two hours after leaving from the same port can be represented with position vectors \vec{OA} [20, 50, 0] and (\vec{OB}) [60, 10, 0]. Assume that the port is located at the origin and that all units are in kilometres. a. How...

I realize by now I must be making everyone crazy on this forum with my questions on vector calculus...but I really have no choice! Please bear with me for another few days...I promise I'll get this done as fast and as painlessly as possible. Here goes... 1.In divergence theorem,green's...

Evaluate the line integral Force field is the integral in the form of integrand ( (2x dx+ 2y dy + 2 zdz)/r^2). the domain of integral is C, C = C1 + C2. C1 is the line segment from (1; 2; 5) to (2; 3; 3). C2,arc of the circle with radius 2 and centre (2; 3; 1) in the plane x = 2. The...

My course is supposed to go from basic vector algebra to extrema, curl and divergence, (stopping before multiple integration) I don't need any help on this assignment I am just wondering how hard it looks compared to everything else. http://tinypic.com/r/2ly29u8/6

Can u suggest a me a really good book on vector calculus and analysis , the book should be more of conceptual nature , because i want to know the indepth concept and meaning of various things like curl, grad, div, various theorems. I mean book shouldn't be problem oriented. I don't need too much...

Homework Statement I need to prove the identity div (a x b) = b dot (curl a) - a dot (curl b) The Attempt at a Solution I've done the proof about 10 times now, and everytime I get the left hand of the identity equal to this: (all the d's are partial derivatives) d(a3b1)/dx -...

I'm not asking for a solution to this problem. I'm just wondering about its validity. Homework Statement The wind velocity v1 is 40mph from east to west while an airplane travels with air speed v2 of 100mph due north. The speed of the airplane relative to the ground is the vector sum v1...

Homework Statement I'm a bit confused as to the following vector calculus identity: [∇ (∇.A)]_i = (δ/δx_i )( δA_j/δx_j) Shouldn’t it be = (δ/δx_i )( δA_i/δx_i) why is it ‘j’ if we are taking it over ‘i’ ? Thanks.

Homework Statement For arbitrary vector fields A and B show that: ∇.(A ∧ B) = B.(∇∧A) - A.(∇∧B) The Attempt at a Solution I considered only the 'i'-axis, by saying that it is perpendicular with A and B and then I expanded both the left and right side out. The working is...

I'm unsure how to do this problem: (a + 2b)\nabla(\nabla \cdot \vec u) - b \nabla \times \nabla \times \vec u - (3a + 2b)c\nabla T(r)= \vec 0 \hat u = U_r \hat r + u_\theta \hat \theta +u_z \hat z a,b,c constants how would I solve this for u?

Homework Statement I need help proving how you could use evaluation of the surface integral \oint\oint f(x,y,z)dS to show that the surface area of the upper hemisphere of radius a is 2\pi a2. So any ideas? Homework Equations The teacher mentioned that the divergence theorem would be...

Recommend a "Vector Calculus" book - by "Vector Calculus" i do not mean Calc III (multivariate). What i mean is the Calculus course that follows after Calc III and consists of topics such as Stokes's and divergence theorems, Cartesian tensors. - Wel'll be using Vector Calculus by Jerrold E...

Homework Statement Prove \int_{V}\nabla\ T d\tau\ = \oint_{S}Td\vec{a} Homework Equations Divergence theorem: \int_{V}(\nabla\bullet\vec{A})d\tau\ = \oint_{S}\vec{A}\bullet\ d\vec{a} The Attempt at a Solution By using the divergence theorem with the product rule for...

Can any show me how you will go about proofing this identity ∇×(u×v)=v∙∇u-u∙∇v+u∇∙v-v ∇∙u where v and u are vectors Many thanks.

Homework Statement Consider a hill that can be modeled by the function z=\frac{14-x^2-y^2}{3} where the +ve x-axis points south and the +ve y-axis points east. Person A is standing at P=(1,2,3) on the hill. i) if A walks due west from P, does he ascend or descend the hill and at what...

Homework Statement Sketch the cone z2=x2+y2 in 3D space. Let (x0,y0,z0)≠(0,0,0) be a point on the given cone. By expressing the fiven equation of the cone in the form f(x,y,z)=a, find a normal vector tot he cone at point (x0,y0,z0) Find the equation of the tangent plane to the cone at...

Homework Statement Let f(x,y,z) be a function of three variables and G(x,y,z) be a vector field defined in 3D space. Prove the identity: div(fG)= f*div(G)+G*grad(f) Homework Equations For F=Pi +Qj+Rk div(F)=dF/dx + dQ/dy + dR/dz grad(F)=dF/dx i + dQ/dy j + dR/dz k The...

Homework Statement Attached Homework Equations The Attempt at a Solution I'm not quite sure how to start these. Once I figure out one of them, I should be able to do the rest. The confusing piece to me is with the unit directional vector (a_rho etc. My teacher has me horribly...

Why is the divergence operation called the 'divergence?' What is the significance of this operation on a vector-valued function? And what about "the curl?" The curl seems self-explanatory (at least it does in electrodynamics), but I need someone to expound on 'the curl' as well.

Homework Statement I am just confused about how the question is structured, and I am unsure on how to get the relevant information to answer the question: Find the angle between the surface normal directions of r^{2} = 9 and x + y + z^{2} = 1 at the joined point (2,-2,1) Homework...

Homework Statement Any sites you recommend that can really supplement my understanding of Vector Calculus applied to Electromagnetis? Thanks

Hello. How can I prove something like \nabla\cdot(\mathbf fv)=(\nabla v)\cdot\mathbf f+v(\nabla\cdot \mathbf f) using only the definition of divergence \text{div}\mathbf V=\lim_{\Delta v\rightarrow0}\frac{\oint_S\mathbf V\cdot d\mathbf s}{\Delta v}, i.e. without referring to...

Hello everyone, I wonder if a textbook like the one I described in the subject line exists - most of the classical electrodynamics textbooks I've looked at assume you already have a solid background in vector calculus. I'm trying to do a self-study in electrodynamics, and while I have taken...

in 2d, "curl"(grad(f)) = 0 "curl" is the operation that green's tehorem talks about. In 3D, curl(grad(f)) = 0 and div(curl(F)) = 0. We may consider vector calculus in 4 spatial dimensions, for vector fields F:R^4 -> R^4. what is "curl" like in 4D, since curl is actually only difined in 3D...

Hello, I'm working through some problems in the Griffith text on electrodynamics. In one of them, the reader is asked to prove the following identity (which is given in the text), which is a generalization (of sorts) on the divergence theorem: \large{ \int_V \left(\nabla T\right) dV =...

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...

Vector Calculus: Surface Integrals Homework Statement Find the surface integral of u[/B dot n over S where S is part of the surface z = x + y^2 with z < 0 and x > -1, u is the vector field u = (2y,x -1,0) and n has a negative z component Homework Equations The Attempt at a...

is it true that if \nabla \cdot \vec{F}=0 , \nabla \times \vec{F}=0 then \vec{F}=0?

My vector calculus is a bit rusty. Can anyone tell me if the following uses proper symbolism? F &= \left[\begin{matrix}f_1(x_1,x_2) \\ f_2(x_1,x_2) \\ f_3(x_1,x_2) \end{matrix}\right] \qquad x = \left[\begin{matrix} x_1 \\ x_2 \end{matrix}\right] \qquad \frac{DF}{dx}&=...

Homework Statement Let r be a position vector from the origin (r=xi+yj+zk), whose magnitude is r, and let f(r) be a scalar function of r. Sketch the field lines of f(r)r 2. Homework Equations 1 \nablax(\nabla\Psi)=0 2 \nabla.(\nablaxv)=0 3...

use green lemma to evaluate line integral int of (x^2-y^2)dx-4xydy over rgn bounded by y^2=4x,y=0,x=1 in ist quadrant. attempt double integral (dv/dx -du/dy)dx dy=int. from limit 0to 2(int.from 0to 2x^1/2(6ydxdy)) =int.lim 0to 2(12x^3/2)dy =24/5(2^5/2) ans in book is -2.

Hi all, Would someone please re-enlighten me. Say I have a vector in spherical coordinates: \vec r_1 = \phi \hat{\phi} + \theta \hat{\theta} + R \hat{R} Where r, \theta, R are scalars and the corresponding hat notation is the unit vectors. Say, I form a new vector r_2 in...

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