Vector calculus Definition and 422 Threads

Homework Statement If \phi(x,y,z) = x3 + 2xy +yz3 find \nabla\phi at the point P=(1,1,2) and direction of the unit normal to the surface \phi(x,y,z) = 11 at P. Homework Equations The Attempt at a Solution Worked out \nabla\phi to be 5i + 10j + 12k Got |\nabla\phi|= √256 so the...

Homework Statement Find a function f(x,y,z) such that F = (gradient of F). The Attempt at a Solution I don't know :( I'm so confused Please help me!

Let c(t )=(2t,sint,cost) be a path. Describe the shape and orientation of this path Describe the shape and orientation between points (0,0,1) and (pi,1,0) I have no idea how to figure out the shape of a curve from its path and my book is only confusing me. Please help!

I'm studying Vector Calculus right now, and I'll have a test about Coordinate Transformation soon. But the book my teacher recommended (Mathematical Methods for Physicists - Arfken) is way too hard to understand this subject. Does anyone know any good material about this that I can find on...

Homework Statement Calculate the line integral: f(x,y) = (x² - 2xy)î + (y² - 2xy)j, between the points (-1,1) and (1,1) along the parabola y = x². (resp: -14/15) The attempt at a solution I thought something like this: substitue y = x², and then integrate de f(x,y). And then evaluate...

Homework Statement \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Equations \vec{r} = x_{i}e_{i} The Attempt at a Solution \frac{∂x_{i}}{∂x_{j}} = 1 iff i=j δ_{ij} = 1 iff i=j therefore \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Statement r^{2} = x_{k}x_{k} Homework...

Homework Statement show that n(t)=-g'(t)i+f'(t)j and -n(t)=g'(t)i-f'(t)j are both normal to the curve r(t)=f(t)i+g(t)j at the point (f(t),g(t)). Homework Equations The Attempt at a Solution i tried finding the unit normal of r(t) in hopes that it would be exactly what n(t) and...

Hi, I am not fully understanding how to express velocity as a vector equation. In an example my professor gave in class, a particle moves (in a straight line) from point A to point B (starting from t=0) and traveling at 6 m/s. The solution included a unit vector. Does that mean all vector...

I need to prove this: u x (\nabla x u) = \frac{1}{2}\nabla(u²) - (u \cdot \nabla)u. I've came to this: uj∂iuj - uj∂jui (i think it's correct) But how this 1/2 appears?

Homework Statement My problem is one pertaining to my Vector Calculus course. The assignment is asking us to "Find equations for the planes tangent to z = x2 + 6x + y3 that are parallel to the plane 4x − 12y + z = 7." The problem I'm having with the problem is the plural aspect. It states...

Homework Statement Calculate the following expressions: Homework Equations The Attempt at a Solution Letting the vector a = (a_1, a_2, a_3) I've worked out that it's 2|a|^2 While that method is fairly quick, I don't particularly like it, and was wondering if there is a shorter or neater...

Homework Statement Use the given information to find the position and velocity vectors of the particle. a(t) = −cos t i − sin t j; v(0) = i; r(0) = j Homework Equations The Attempt at a Solution Ok first step integrate a(t). which i get to be -sin(t)i +cos(t)j + c now...

Homework Statement I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce. Homework Equations \vec{e} \cdot \frac{\partial}{\partial t}(\rho \vec{u}) = -\nabla\cdot (\rho\vec{u})\cdot\vec{e} -...

Besides Apostol 2, is there any good, rigorous and suited for self study book on this subject? Thanks

Homework Statement So using standard spherical polar co-ordinates, my notes define a sphere as r(s,t) = aCos(s)Sin(t) i + aSin(s)Sin(t) j + aCos(t) k and the normal to the surface is given by the cross product of the two partial differentials: \partialr/\partials X \partialr/dt...

Homework Statement See the photo on the left Homework Equations The Attempt at a Solution

Looking for supplemental material. This is the textbook I am supposed to use. (http://www.amazon.com/dp/0321549287/?tag=pfamazon01-20) The syllabus is: Vector fields; vector calculus; ordinary differential equations; sequences, series, and power series.

Homework Statement With \vec{L} = -i\vec{r} x \nabla, verify the operator identities \nabla = \hat{r}\frac{\partial }{\partial \vec{r}}-i\frac{\vec{r}\times\vec{L}}{r^{2}} and \vec{r} \bigtriangledown ^2 - \bigtriangledown (1+\vec{r}\frac{\partial }{\partial \vec{r}})=i\bigtriangledown \times...

Homework Statement Problem 1: Two fixed unit vectors A and B make an angle θ with each other, where 0 < θ < π. A particle moves in a space curve in such a way that its position vector r(t) and velocity v(t) are related by the equation v(t) = A x r(t). If r(0) = B, prove that the curve has...

so we have the identity \nabla\times\nabla\phi = 0 and from Maxwell's equations we have \nabla\times \textbf{E} = -\frac{d\textbf{B}}{dt} But we also have that \textbf{E} = -\nabla\phi So the problem I'm having is this -\textbf{E} = \nabla\phi which i substitute into the...

ive been trying to understand a few of the identities my professor gave me and i can get a few of them down such as \nabla(\vec{A}\vec{B})=\vec{B}\nabla\vec{A} - \vec{A}\nabla\vec{B} and i can break it down through cartesian and product rules but when i try to do \nabla X (\vec{A}ψ) =...

I am trying to figure out a proof for this identity \nabla(f/g) = (1/g2) (g\nablaf - f\nablag) Any ideas?

Can anyone list some useful tricks for calculus and vector calculus. i.e. sneaky substitutions, changing limits of integration when changing coordinate systems, parametrization of curves, surfaces, and volumes etc.

Homework Statement there is a surface xy3z2=4. What is the unit normal to this surface at a pt in the surface (-1,-1,2)?? Homework Equations what is a unit normal to a scalar region? how can it be calculated? The Attempt at a Solution i calculated the gradient (del operator) of...

Hi, i have just finished self-studying spivak calculus and have thoroughtly enjoyed reading it and doing the problems. I am looking to find a book on vector calculus with similar rigor as that of spivak. Any recommendations? I have heard Hubbards book on vector calculus is not bad. Any...

Hi! Could you recommend a book related with vector calculus? I want to get a book which is written so rigorously that it helps me understand the topic 'vector calculus' more precisely. I'll wait for your good recommendations. Thanks!

Homework Statement (Sorry for the confusing font- I tried to figure out using the latex reference to indicate a vector, but couldn't do it- please view the "superscripted" variable as a vector. Thanks) A. The electrostatic potential at P arising form a dipole of unit strength at O...

1.10.1 The force field acting on a two-dimensional linear oscillator may be described by F=−ˆxkx − ˆyky. Compare the work done moving against this force field when going from (1, 1) to (4, 4) by the following straight-line paths: (a) (1, 1)→(4, 1)→(4, 4) (b) (1, 1)→(1, 4)→(4, 4) (c) (1...

How to get U(r) and W(r), why are them defined out of nowhere, is that the standard structure of that kind of functions ( what kind of functions are them??) ? T.Y

Homework Statement Prove \nabla \bullet (\textbf{A} \times \textbf{B}) = \textbf{B} \bullet (\nabla \times \textbf{A}) - \textbf{A} \bullet (\nabla \times \textbf{B}) I'd like to prove this using the levi-civitia symbol: \epsilon_{ijk} and einstein-summation convention as practice and...

Can someone help me prove the identity \ u \times (\nabla \times u) = \nabla(u^2 /2) - (u.\nabla)u without having to write it out in components?

Homework Statement Q1. Evaluate grad(f) for the function f(\underline{r})=(\underline{a} dot \underline{r}) (\underline{b} dot \underline{r}) Q2. If \underline{c} is a constant vector, show that grad |\underline{c} cross \underline{r}| ^n = n |\underline{c} cross \underline{r}| ^(n-2)...

Ok, the professor I got who is supposed to teach Vector Calculus and Differential Equations sucks and also does the book he uses. I am a Mechanical Engineering student and I will begin self teaching those subjects. For that matter, I would like good books, if possible. I sense those subjects are...

How to prove that cyclic integral of (r dot ds) =0 (symbols having usual meanings). Please help me 2) It is always possible to find curl when vector function is known, but how to find the vector when its curl is known.

Homework Statement Hi, this is not part of a problem but just an equation I'm having a hard time to decipher (for the reference the original one is in "Statistical Mechanics" by Pathria, eq. 3.7.16) We define: r = |r2–r1|, Where bold letters are vectors, and we basically integrate a function...

I'm manipulating an equation, and I think I am correct in doing this, but not sure. Could someone tell me if the equality I've written below is true? [\nabla\cdot [\rho\vec{v}\vec{v}] ]\cdot\vec{v} = \nabla\cdot[\frac{1}{2}\rho v^2 \vec{v}] (where \rho is dependent on position) *NOTE* that...

I'm working on simplifying a big physical expression (I don't like the Navier-Stokes equations at all anymore), and I'm curious how to simplify the following term: \vec{v}\cdot (\vec{v}\cdot\nabla )\vec{v} where v is a fluid velocity - i.e. definitely spatially varying. I'm just not sure...

Although I am an aspiring physicist, I cannot cope with the physicist's love for vagueness when it comes to yielding math. Exactness is simply not a luxury that can be ignored, certainly not in theoretical physics. But okay, I realize the dirac delta function can be made exact by the use of...

Homework Statement [PLAIN]http://img576.imageshack.us/img576/1710/vectorp.png Homework Equations The Attempt at a Solution I've done part (a) but how do I do (b)?

Homework Statement [PLAIN]http://img585.imageshack.us/img585/526/indexnotation.jpg The Attempt at a Solution How do I proceed?

I'm in high school, and right now I'm taking AP Calculus. I'm interested in dual enrolling at a Community College over the summer. I've been looking at their Math selection, and they list Vector Calculus as a course. It sounds, from reading the...

The question is Let r=r(x,y,z) where it is the distance from a point O. Evaluate \oint\nabla(1/r)*ndS (where * is the dot product) over the ellipsoid x^2/4 +y^2/9+z^2/25=1 I thought the answer was 0 since the ellipsoid is a simply connected region in R^3 and the...

Homework Statement Question 3 part b and c Homework Equations Divergence and Stokes Theorems. Knowledge of parametrization ect ect The Attempt at a Solution I got the B field by using curl. However any attempt to resolve the flux through the top hemisphere or even the...

Can anybody please suggest me a good book which covers following topics in detail Vectors - Dots, Cross and triple products, Gradient, divergence and applications. thnx in advance

Homework Statement The Force on a mass with position vector r satisfies: m\frac{d^{2}\textbf{r}}{dt^{2}}=F=f(\textbf{r})\textbf{r} where f(r) is scalar function of r. Show that L: L=\textbf{r}\times\frac{d\textbf{r}}{dt} is conserved. Homework Equations The Attempt at a...

Homework Statement Let g: D-->R. D subset of R^3 be harmonic Then for any closed ball that is a subset of D with radius >0 and its origin in D With its surface s=(partial d)B(a,r) = {x={x1,x2,x3} st mod(x-a) = r} show that f(a)=(1/4.pi.r^2)INTRGL: f (over s) Homework Equations...

I'm reading Marsden's vector calculus. In the chapter of differential forms, it mentions the wedge product satisfies the laws: dy^dx=-dxdy. and for a 0-form f, f^w=fw. Does it have formal derivation? hope someone can give me a hint or even a link.

Homework Statement find the area if the vertices are (3,9,8),(0,5,1),(-1,-3,-3),(2,1,4) Homework Equations The Attempt at a Solution I draw the points and I couldn't know the shape it is complex I really couldn't know it

Homework Statement Hello. I want to see if I am interpreting the following correctly, I certainly don't expect anyone to work the problem out as it is (at least with my approach) fairly tedious. Compute the following: (\vec{V}\cdot\nabla)\vec{U} (\vec{U}\cdot\nabla)\vec{V} Given: \vec{r} =...

Hello, I'm learning vector calculus in my physics class. We're using Mathematical methods for Physicists by Arfken, but I think it makes a better reference source than something to learn the concepts from. I've downloaded the Feynman lectures which seem to be pretty good so far, but I was...