Vector field Definition and 382 Threads

Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous? How can I prove it? Thanks.

Homework Statement y'=ay-by^2-q, where a, b are positive constants, and q is an arbitrary constant. In the following, y denotes a solution of this equation that satisfies the initial condition y(0) = y_0. a. Choose a and b positive and q < a^2/4b. By plotting direction fields and...

My related questions 1 Is there any difference between 'vector field' and 'vector function'? 'vector function' is also called 'vector-valued function' (Thomas calculus). According to their definitions, they are all the same things to me. And they are all some kind of mapping, which assigns a...

Homework Statement A vector field is defined by F(x) = (y+z, x+y, x+z). Find the Jacobian and determine if the field is conservative in a finite region. If it is conservative, find the potential function. Homework Equations F = delta p AKA F = (upsidedown triangle) p The Attempt...

In physics one often uses the following: If the rotation of a vector field A vanishes, one can write A as the gradient of some scalar field, i.e. rot(A)=0 \Rightarrow A=\bigtriangledown \Phi. Is this true without further restrictions? If yes: Why? Thanks in advance...Cliowa

Homework Statement A vector field V is not irrotational.Show that it is always possible to find f such that fV is irrotational. Homework Equations The Attempt at a Solution \nablax[fV]=f\nablaxV-Vx\nablaf I have to equate the LHS to zero.But then,how can I extract f out of the...

To show that a vector field F=(P,Q,R) is conservative, is it enough to show that DP/DY = DQ/DX?

Hello everyone I'm not sure if this is right or not... If i have F(x,y,z) = zj; where j is the vector, j hat. Would that be all vectors are going to be pointing up if you assume z is up, and are in the y plane? If the coordinate system is, z is up, y is to the right, and x is...

When we say condition of a vector field F being conservative is curl F=0,does it mean that F=F(r)?.I know normally it does not look so.Please,then site an example where F is not a function of r,but still curl F=0.

Just a quick question about notation. I was given the vector field F = r + grad(1/bar(r)) where r= (x)i+(y)j+(z)k. grad is just written as the upside down delta (gradient) and the bar I wrote in the above equation looks like an absolute value around just the r (although I don't know if it...

Homework Statement If the divergence of a vector field is zero, I know that that means that it is the curl of some vector. How do I find that vector? Homework Equations Just the equations for divergence and curl. In TeX: \nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial...

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...

Hi Guys, Given the vector field X(x,y) = ( a + \frac{b(y^2-x^2)}{(x^2+y^2)^2}, \frac{-2bxy}{(x^2+y^2)^2}}}) Show that for a point (x,y) on the circle with radius r = \sqrt(b/a) (i.e. x^2 + y^2 = b/a), the vector X(x,y) is tangent to a circle at the point. My strategy is that to first...

Again, I'm stuck on a question: "Let C be the region in space given by 0 \leq x,y,z \leq 1 and let \partial C be the boundary of C oriented by the outward pointing unit normal. Suppose that v is the vector field given by v = (y^3 -2xy, y^2+3y+2zy, z-z^2) . Evaluate \int_{\partial...

Let S be the part of the plane 3x+y+z=4 which lies in the first octant, oriented upward. Find the flux of the vector field F=4i+2j+3k across the surface S. \int \int F\cdot dS = \int int \left( -P \frac{\partial g}{\partial x} -Q \frac{\partial g}{\partial y} +R \right) dA \int \int \left(...

Alright, so the field is \mathbf{F} = (z^2 + 2xy,x^2,2xz) it's a gradient only when f_x = z^2 + 2xy, f_y = x^2 and f_z = 2xz integrate the first equation with respect to x to get f(x,y,z) = \int z^2 +2xy\,dx = xz^2 + x^2y + g(y,z) now, f_z(x,y,z) = g_z(y,x) which is 2xz integrate that with...

Please could someone explain to me how to visualise a vector field? Let's say it's v(x, y) = (2.5, -x) on whatever domain. I tried it the same way as I would visualize a scalar field but the results did not correspondent at all with the results I'd expect. The same for drawing the field along...

Is there a simple way to express a general vector field in terms of the gradient of another (perhaps higher dimensional) function?

The question should be very easy, its from topics of Differential Geometry, I just want to make sure that I understands it right :shy: . My question is: in R^3 we have vector field X and for every point p(x,y,z) in R^3 space, vector field X(p) = (p; X_x(p), X_y(p), X_z(p)) has: X_x(p) =...

Question: Vector field and (n-1)-form representation of current density Electric current density can be represented by both a vector field and by a 2-form. Integrating them on a given surface must lead the same result. My question is, what is the relation between this vector field and the...

I am terribly sorry for not being able to write this simple equation in Latex form. (I will be really glad if someone can tell me where I can learn how to use Latex to write math symbol) Let F' be a vector field given by F' = r r' (r' = radial unit vector) and also let p be a point on the...

Should your answer include the constant of integration? I think it should but my book's answers don't, so I dunno. Example, <2xy^3, 3y^2x^2> answer is x^2y^3, but should I include the + C? (and yes I went through and made sure h(y) was in fact a constant

A vector field is difined by A = f(r)r. a) show that f(r) = constant/r^3 if divergence A equal to zero. b) show that curl A is alway equal to zero

Given a vector field: \vec F= (x^2-xy)\hat x +(y^2-yz)\hat y +(z^2-xz)\hat z Find the conditon for the divergence to be equal to zero.

Something we did in electrostatics that's a source of confusion for me: We learned to use caution when taking the divergence of the (all important) radial vector field: \vec{v} = \frac{1}{r^2} \hat{r} Applying the formula in spherical coords gave zero...a perplexing result. The...

Hey ya'll, How do I find the potential function of this conservative vector field (It is conservative isn't it?? I did check, but i might've messed that up too!). \int (2x-3y-1)dx - (3x+y-5)dy I know to break the function: F(x,y)= (2x-3y-1)i - (3x+y-5)j apart and integrate...

If the curl of a vector field is zero, then we can that the vector field is path independent. But there are cases where this is not true, I was wondering how? Whats the explanation for this? Thanks in advance for any help. - harsh

Hi there Can somebody please explain shortly the difference between a tangent vector and a vector field? I'm still new to differential geometry. I read couple of sources that had mixed claims on which of them actually act on a given function f. so I'm kind of confused. Much appreciated.

What are they? "A fundamental property of the natural world that is of supreme importance for physics. It has two components: rotational symmetry, and boost symmetry." :confused:

ok this probley seems simple but i just need to see how to do it, ok well how do u evaluate this... find the flux of the vector field... \vec{F}=<x,y,z> throught this surface above the xy-plane.. z = 4-x^2-y^2 how do u evaluate this with surface integrals method and the divergence...

I was just wondering; how is a vector valued function different from a vector field? Mathematically, they seem the same so should I think of them that way?

Viva! I usually come upon this statement: " Since B is solenoidal, it can be split into Toroidal and Poloidal parts, i.e, B=Bt+Bp, where Bt=curl(Tr) and Bp=curlcurl(Pr)" How can I prove this?? I think it is somehow related with the stokes theorem... Looking forward for...