Vector fields Definition and 161 Threads

Hi all. I have difficulty in visualizing the concept of divergence of a vector field. While I have some clue in undertanding, in fluid mechanics, that the divergence of velocity represent the net flux of a point, but I find no clue why the divergence of an electric field measures the charge...

Comment: My question is more of a conceptual 'why do we do this' rather than a technical 'how do we do this.' Homework Statement Given a lie group G parameterized by x_1, ... x_n, give a basis of left-invariant vector fields. Homework Equations We have a basis for the vector fields...

i am new in using MAPLE. can anyone please tell me how i can draw a vector field in MAPLE? suppose i want to draw the vector field of \vec{A} = x\hat{i} + y\hat{j}. or you can take any other vector you like to demostrate how i can draw it.

I am learning vector analysis these days. I have some knowledge of the applications of the vector field in the field of Engineering and Astrophysics. I am also aware of the fact that vector field is a function that assigns a unique vector to each point in two or three dimensional space...

I need help solving the following problem: Let M,N be differentiable manifolds, and f\in C^\infty(M,N). We say that the fields X\in \mathfrak{X}(M) and Y \in \mathfrak{X}(N) are f-related if and only if f_{*p}(X(p))=Y_{f(p)} for all p\in M. Prove that: (a) X and Y are f-related if and only if...

I'm self-studying Schutz Geometric methods of mathematical physics, having problems with ex. 2.1. page 44. r=cos(theta)x+sin(theta)y theta=-sin(theta)x+cos(theta)y show this is non-coordinate basis, i.e. show commutator non-zero. I try to apply his formula 2.7, assuming...

I'm self-studying Schutz Geometric methods of mathematical physics, having problems with ex. 2.1. page 44. r=cos(theta)x+sin(theta)y theta=-sin(theta)x+cos(theta)y show this is non-coordinate basis, i.e. show commutator non-zero. I try to apply his formula 2.7, assuming...

Hi, I'm having trouble with the following question. Q. Let p be a real constant and \mathop F\limits^ \to = \left( {yz^p ,x^p z,xy^p } \right) be a vector field. For what value of p is the line integral \int\limits_{C_2 }^{} {\mathop F\limits^ \to \bullet d\mathop s\limits^ \to } =...

I want to find which values of n make the vector field \underline{F} = {|\underline{r}|}^n\underline{r} solenoidal. So I have to evaluate the divergence of this vector field I think, then show for which values of n it is zero? Im starting by substituting: \underline{r} = \sqrt{x^2...

Suppose we have a conservative vector field on a plane. Suppose also that we have a closed curve C on that plane. Then we have: \int_C \mathbf{F}\cdot d\mathbf{r} = 0 The line integral around C is zero because F is conservative. Here is what I don't understand: If you have one or more...

I recall an prof of mine (from a grad EM course) telling me that once the divergence and curl are of a vector field is specified then the vector field is determined up to an additive constant. Helmhotz's Theorem states that any vector field whose divergence and curl vanish at infinity can be...