1. ### Differential map between tangent spaces

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...
2. ### Question about conditions for conservative field

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...
3. ### How to compute the surface height based on normal vectors

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 =...
4. ### Can I pull a time derivative outside of a curl?

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?
5. ### Divergence Theorem Question (Gauss' Law?)

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...
6. ### Construct B field from a given E field using Maxwell's Eqns

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...
7. ### Calculus Regarding to Multi-Variable Calculus Books

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...
8. ### Intuitive interpretation of some vector-dif-calc identities

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...
9. ### Calculating Flux through Ellipsoid

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 ##...
10. ### Plotting surfaces in 3-space

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...
11. ### Equation of the tangent plane in R^4

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...
12. ### Translation from vector calc. notation to index notation

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...
13. ### Vector Calculus - Use of Identities

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...
14. ### Verification of Stoke's Theorem for a Cylinder

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]...
15. ### Most rigorous book covering Gauss, Green & Stokes theorems,++ -Any recommendations

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...
16. ### Proof of equivalence between nabla form and integral form of Divergence

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