Vector calculus Definition and 416 Threads

Hello everyone, How far can the book take me before I have to read another book on the subject? Thanks :)

1. The problem is: ( a x b )⋅[( b x c ) x ( c x a )] = [a,b,c]^2 = [ a⋅( b x c )]^2 I am supposed to solve this using index notation... and I am having some problems. 2. Homework Equations : I guess I just don't understand the finer points of index notation. Every time I think I am getting...

Dear Physics Forum advisers, Could you recommend books that treat the multivariable calculus from a theoretical aspect (and applications too, if possible)? I have been reading Rudin's PMA and Apostol's Mathematical Analysis, but their treatment of vector calculus is very confusing and not...

Is there a way to simplify the proof of different vecot calculus identities, such as grad of f*g, which is expandable. And also curl of the curl of a field. Is there a more convenient way to go about proving these relations than to go through the long calculations of actually performing the curl...

Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it. Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result: \vec...

Homework Statement I want to know if I got the answer correct and if my reasoning is sound. The text answers and solutions manual only gives answers/solutions for odd numbered problems. Here is the problem: And a direct link to the imgur page: http://i.imgur.com/Tko1xFh.png Homework...

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...

Homework Statement [/B] A swimmer located at point A needs to reach a point B 20 meters downstream on the opposite bank of a 10 meter wide river. The river flows horizontally at a rate of 0.5 meters/second, and the swimmer has a constant speed of 0.25 meters/second. Set up the vector...

I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...

I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks. Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2) at point 1,2 a) in which...

Question about conditions for conservative field In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open. Usually in textbooks there is not much explanations on why these...

Suppose I have already found the surface normal vectors to a set of points (x,y), how do I compute the surface height z(x,y)? Basically what I have are the normal vectors at each point (x,y) on a square grid. Then I calculate the vectors u = (x+1,y,z(x+1,y)) - (x,y,z(x,y)) and v =...

Homework Statement For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get ∇ x (∇ x E) = ∇ x -∂B/∂t I feel like it'd be very wrong to pull out the time derivative. Am I correct?

Are there fields of "pure" Calculus that follow Vector Calculus? I mean fields that are not primarily ODE, PDE etc, Real Analysis or Complex Analysis, topology etc. Does Vector Calculus and its prerequisites fully encompass the basics of Calculus as a field?

If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0 I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...

Homework Statement Given an electric field in a vacuum: E(t,r) = (E0/c) (0 , 0 , y/t2) use Maxwell's equations to determine B(t,r) which satisfies the boundary condition B -> 0 as t -> ∞ Homework Equations The problem is in a vacuum so in the conventional notation J = 0 and ρ = 0 (current...

Dear Physics Forum personnel, I am a college sophomore with double majors in mathematics & microbiology and an aspiring analytic number theorist. I will be going to self-study the vector calculus by using Hubbard/Hubbard as a main text and Serge Lang as a supplement to Hubbard; this will help...

Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...

Homework Statement Let ## E ## be the ellipsoid: $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$ Let ## S ## be the part of the surface of ## E ## defined by: $$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$ Let F be the vector field defined by $$ F=(-y,x,0)$$ A) Explain why ##...

Homework Statement Given the eqn x=2, y=sin(t), z=cos(t), draw this function in 3-space. Homework Equations ABOVE^ The Attempt at a Solution I did this: x^2+y^2+z^2=2^2+(sin(t))^2+(cos(t))^2=5 Therefore we get x^2+y^2+z^2=5 Which is the eqn of a sphere with radius root5. My friend said it's...

Let f: \mathbb R^2 \to \mathbb R^2 given by f=(sin(x-y),cos(x+y)) : find the equation of the tangent plane to the graph of the function in \mathbb R^4 at (\frac{\pi}{4}, \frac{\pi}{4}, 0 ,0 ) and then find a parametric representation of the equation of the tangent plane What I did: the...

THIS THREAD HAS BEEN MOVED FROM ANOTHER FORUM BECAUSE IT IS HOMEWORK. BUT THERE IS NO TEMPLATE. Hi everyone, The problem gives that a particle moves in a circle with angular velocity ω. I know that r×ω=v, which is the velocity of the particle. However, I am told to differentiate and find τ=Iα...

Homework Statement Air is flowing with a speed v in the direction (-1, -1, 1,) calculate the volume of air flowing through the loop consisting of straight lines joining (in order i presume) (1,1,0) (1,0,0) (0,0,0) (0,1,1) (1,1,0) Homework EquationsThe Attempt at a Solution I assume you have to...

Homework Statement The line L L1: x=3+2t, y=2t, z=t Intersects the plane x+3y-z=-4 at a point P. Find a set of parametric equations for the line in the same plane that goes through P and is perpendicular to L. Homework Equations cross-product r=r0+t(vector) this is to get in parametric form...

Hi, I want to translate this equation R_{\hat{n}}(\alpha)\vec{x}=\hat{n}(\hat{n}\cdot\vec{x})+\cos\left(\alpha\right)(\hat{n}\times\vec{x})\times\hat{n}+\sin\left(\alpha\right)(\hat{n}\times\vec{x}) to index notation (forget about covariant and contravariant indices). My attempt...

Homework Statement I need to find a sequence of partitions , let's call it S of R=[0,1]x[0,1] such that as the number of partitions k→∞ , then limit of the area of the largest subinterval of the rectangle in the partition, denoted a(S) tends to 0, but the mesh size m(S) is a non-zero value...

Homework Statement Let R be the unit square such that R= [0,1] x [0,1] Find a sequence of partitions of R such that the limit as k ->inf of the area of the largest sub-rectangle of the partition (where k is number of partitions) goes to zero but the mesh size does not go to zero. Depicting the...

Hi, I need help getting a start on this exercise. Let R be the unit square. Find sequence of partitions such that the mesh size goes to zero as sequence goes to infinity -> for this, is this just a series of sub rectangles whose respective areas shrink at the same rate with respect to one...

Hi. I was looking for a chain rule in vector calculus for taking the gradient of a function such as f(A), where A is a vector and f is a scalar function. I found the following expression on wikipedia, but I don't understand it. It's taking the gradient of f, and applying that to A, and then...

Homework Statement By using a suitable vector identity for ∇ × (φA), where φ(r) is a scalar field and A(r) is a vector field, show that ∇ × (φ∇φ) = 0, where φ(r) is any scalar field. Homework Equations ∇×(φA) = (∇φ)×A+φ(∇×A)? The Attempt at a Solution I honestly have no idea how to even...

I am in Calculus BC in high school right now and I am really enjoying it and have finished all of the material for the year and I have heard different things about what class comes after Calculus BC (I believe BC is equivalent to Calc 1 & 2 in college). I have heard the next class, aka Calc 3...

I have a question, to use cylindrical coordinates to find the volume of the ellipsoid ${R}^{2}+{3z}^{2}=1$. I know for cylindrical coordinates the Jacobian is $r$ so I have some integral: $$\iiint (r)dzdrd\theta$$ However I am struggling to work out the bounds of the integral for $z,r,\theta$...

Homework Statement Homework Equations Stoke's Theorem: The Attempt at a Solution ∇×A = (3x,-y,-2(z+y)) I have parametric equation for wall and bottom: Wall: x(θ,z) = acosθ ; y(θ,z) = asinθ ; z(θ,z) = z [0≤θ≤2π],[0≤z≤h] Bottom: x(θ,r) = rcosθ ; y(θ,r) = rsinθ ; z(θ,r) = 0 [0≤θ≤2π],[0≤r≤a]...

Hi. Every time i read physics I feel so discouraged by my lack of understanding in terms of vector calculus and trigonometry. I do understand basic trigonometry and basic vectors but when professors or textbooks move onto combining derivatives with vectors i lack understanding of which rules...

Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables. My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for...

Problem: Consider a system for which Newton's second law is $$ \frac {d \vec v}{dt} = - [ \frac {h(r)h'(r)}{r} + \frac {k}{r^3} ] \vec r- \frac {h'(r)}{r} \vec L $$ where k is a constant, h(r) is some function of r, h'(r) is its derivative and L = r x v is the angular momentum. Show that $$...

Does anybody knows how you can reach one form of the divergence formula from the other? Or in general, why is the equivalence true?

I have attached an image... Okay, so I have been stuck on this problem for like 2 hours now and I have no idea how to find r(x). I know the trace is the intersection of the plane and the surface. My first attempt was to substitute the plane y+2x=0 equation for the surface equation by solving...

Homework Statement div(øu) = ødivu + ugradø Homework Equations divergence of scalar field = f,ii divergence of vector field = ui,i The Attempt at a Solution I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm...

Homework Statement let r (vector) =xi+yj+zk and r=sqrt(x^2+y^2+z^2), let f(r) be a C2 scalar function. 1. Prove that ∇f = dr/df * vector r 2. Using part 1, calculate ∇ cosh(r^5), check answer by direct calculation 3. Using Vector Identities, calculate ∇ X (cosh(r^5)*∇f...

I'm doing an undergraduate course in engineering 1st sem. I have physics which contains cumbersome amount of vector calculus operations. Although I have cleared my concepts I require a lot of practice to increase agility in vector calculus. I demand thus a good practice book or set of problems...

Hi guys. I hope this isn't a bad place to post my question, which is: I'm reading some lecture notes on Lagrangian mechanics, and we've just derived the Euler-Lagrange equations of motion for a particle in an electromagnetic field. It reads: m \ddot{\vec{r}} = -\frac{e}{c} \frac{\partial...

Homework Statement (a) Show that the four points r1 = (1, 0, 1), r2 = (4, 3, 5), r3 = (6, 4, 6) and r4 = (3, 1, 2) are coplanar and the vertices of a parallelogram. Let S be the closed planar region given by the interior and boundary of this parallelogram. An arbitrary point of S can be...

A scalar field \psi is dependent only on the distance r = \sqrt{x^{2} + y^{2} + z^{2}} from the origin. Show: \partial_{x}^{2}\psi = \left(\frac{1}{r} - \frac{x^{2}}{r^{3}}\right)\frac{d\psi}{dr} + \frac{x^{2}}{r^{2}}\frac{d^{2}\psi}{dr^{2}} I've used the chain and product rules so...

I must become good at this ASAP. Homework Statement prove \vec{\nabla}\cdot (\vec{a}\times\vec{b} ) = \vec{b} \cdot(\vec\nabla\times\vec{a}) - \vec{a}\cdot(\vec\nabla\times\vec{b}) Homework Equations \vec a \times \vec b = \epsilon_{ijk}\vec a_j \vec b_k \vec\nabla\cdot =...

I've taken a multi-variable calculus course already that covers infinite sequences and series, Taylor's theorem, quadratics surfaces, double and triple integration etc. I'm looking to get a Master's Degree in statistics two years from now, is there any point of me taking a class that involves...

Homework Statement Use your knowledge of vector algebra to verify the following identity: \vec{\Omega} \cdot \nabla n = \nabla \cdot \vec{\Omega} n Homework Equations Divergence product rule \nabla \cdot (\vec{F} \phi) = \nabla (\phi) \cdot \vec{F} + \phi (\nabla \cdot \vec{F})...

The following is used as part of a proof I'm trying to understand: ∫Vf(∇.A)dV=∫SfA.dS-∫VA.(∇f)dV where f is a scalar field, and the surface integral is taken over a closed surface (which presumably encloses the volume). I'm not sure how to go about proving this. I can see the divergence...

Note: physics conventions, \theta is measured from z-axis We have a vector operator \vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right) And apparently \vec{L}\cdot\vec{L}=...

Hellow! I was noting that several definitions are, in actually, expressions of vector calculus, for example: Jacobian: \frac{d\vec{f}}{d\vec{r}}=\begin{bmatrix} \frac{df_1}{dx} & \frac{df_1}{dy} \\ \frac{df_2}{dx} & \frac{df_2}{dy} \\ \end{bmatrix} Hessian: \frac{d^2f}{d\vec{r}^2} =...