What is Jacobian: Definition and 168 Discussions

Hello! I am having problems with the inverse function theorem. In some books it says to be locally inversible: first C1, 2nd Jacobian determinant different from 0 And I saw some books say to be locally inversible, it suffices to change the 2NDto "F'(a) is bijective".. How could these two be...

Is there any study of the problems associated with the use of the inverse jacobian to go from flat space(time) to curved space(time)? I know they use the jacobian in curvilinear coordinates that parameterize flat space to convert the volume element in curved spaces to volume elements in flat...

Let [phi](u,v)=(3u+v,u-2v). Use the Jacobian to determine the area of [phi]R for: R=[2,5]X[1,7] The Attempt at a Solution - I'm really not sure why I keep getting the wrong answer on this problem. the problem gives you two R's to solve for and I got the right answer for the first...

Here is the problem: Suppose that g is a diffeomorphism on R^n. Then we know that its jacobian matrix is everywhere invertible. Let us define the following matrix valued function on R^n H_{i,j} (x) = \int_0^1 \partial_i g^j(tx) dt where g^j are the components of g. Question : Is...

Homework Statement Could you please help me with this problem? Let f(x) = (f_1(x), f_2(x)) map R^{2} into itself where f_1, f_2 have continuous 1st/ 2nd partial derivatives in each variable. Assume that f maps origin to itself and that J_f(x)(Jacobian matrix) is an invertible 2x2 matrix for...

What does a jacobian mean? I know what it IS, as in, if given a set of multivariable functions, I can find out the jacobian, but what does it MEAN? And why do we use it to change between coordinate systems (cartesian-> polar =|jacobian of polar|* function in polar coordinates)?

Is Jacobian Conjecture still open for general case? who knows the recent progress on this problem, especially in the approach of Feyman graph? 3x.

If I am using the method of characteristics to solve a PDE \Psi(x,t) (first order, semi-linear), and after using the method of characteristics I find that the Jacobian |\frac{\partial{(x,t)}}{\partial{(\sigma,\eta)}}| = 0 (where \sigma and \eta are parameters for the curve) does this imply...

So in the change of coordinates equation for multiple integrals, we have the Jacobian which allows us to change our bounds of integration. It's the connection between multivariable calculus and linear algebra. is this possible since the linear transformation allows a one-to-one-mapping...

Does anyone know any reference or proof to the statement that since a flow is irrotational, the Jacobian is symmetric?

jacobian matrix with 2 variables please help! Homework Statement so we have z=x^2+x^3 and z=y+sin(x). Find the jacobian matrix of this system. Find the determinant of this jacobian. The Attempt at a Solution The determinant part is easy, the only problem is trying to set this up. I'm...

Can someone help me with the following? I am supposed to evaluate ∫∫ e^(x+y)dA where the area of integration is given by the inequality |x|+|y|≤1. So, suppose I do one of these Jacobians, and I set u = |x| and v = |y|, so wouldn’t the equation have to satisfy the inequality u+v≤1, and...

Hi, I have the following integral. \int\limits_{}^{} {\int\limits_R^{} {\left( {\sinh ^2 x + \cos ^2 y} \right)} \sinh 2x\sin 2ydxdy} Where R is the part of the region 0 <= x, 0 <= y <= pi/2 bounded by the curves x = 0, y = 0, sinhxcosy = 1 and coshxsiny = 1. In the hints section...

Can someone explain to me this part of the proof of the jacobian? I don't know what they're talking about...I can follow the rest (the cross product bla bla bla bla bla) but I don't know how they're getting these two vectors...I figured it has something to do with partial differentials but I...

Hey everyone, What is the purpose of the Jacobian in change of variables integration? Does it have something to do with the fact that you are basically performing a linear transformation on a set that you are integrating over? There's no rush on this, I was just wondering. Any websites or...

Solve u = 2x - 3y & v = -x + y in terms of x & y, then find Jacobian. Here is the problem: Solve the system u = 2x\;-\;3y,\;\;v = -x\;+\;y for x and y in terms of u and v. Then find the Jacobian \frac{\partial\left(x,\;y\right)}{\partial\left(u,\;v\right)}. Find the image under the...

Evaluate \int\int_{R} \left(2x^2 - xy - y^2\right) dx\;dy by applying the transformation u = x - y , v = 2x + y for the region R in the first quadrant bounded by the lines y = -2x + 4, y = -2x + 7, y = x - 2, y = x + 1 I don't even know where to start! Please help.

Anyone have time to check my Jacobian for this transformation!? x = e^{u-v} y = e^{u+v} z = e^{u+v+w} I ended up getting the Jacobian as ZERO. This is why I am doubting myself--- it seems wrong! What do you guys get? Thanks for you help. :redface: