Jacobian Definition and 163 Threads

Calculus: Coordinate Changes, Jacobian, Double Integrals?? Homework Statement Show that T(u,v) = (u2 - v2, 2uv) maps to the triangle = { (u,v) | 0 ≤ v ≤ u ≤ 3 } to the domain D, bounded by x=0, y=0, and y2 = 324 - 36x. Use T to calculate ∬sqrt(x2+y2) dxdy on the region D...

Hello. When one is converting between coordinate systems, the Jacobian arises as a necessary consequence of the conversion. Does this occur with transformations between relativistic systems, and, if so, is this manifested through the prevalence of gamma in the transforms? Any guidance would...

Hi. First off I don't know if this is the right topic area for this question so I'm sorry if it isn't. So my current situation is that I can find the jacobian matrix for a transformation from spherical to cartesian coordinates and then take the inverse of that matrix to get the mapping from...

Homework Statement Hi I wish to perform an integral of the form \int_0^a {\int_0^b {f\left( {x - y} \right)dxdy} } What I do first is to define s := x-y, and ds = dx. Then we get \int_0^a {\int_{-y}^{b-y} {f\left( {s} \right)dsdy} } Then I can define t := x+y, so dt = dy. Then I get...

in partial differentiation why we have to use the jacobian?what does signifies?how does it differ from normal partial derivative? thanks

Suppose I am changing variables from (x,y) to (s,t), where \begin{align*} s & = \frac 12 (x+y),\\ t & = y - x \end{align*} According to Wikipedia, if I want to see how the measure dx dy changes, I need to compute the Jacobian matrix J associated with this variable transformation and take its...

Homework Statement An elliptical galaxy has gravitational boundaries defiend by 9x^2+16y^2+144z^2=144. A black hole at the center of the galaxy is interacting with dark matter producing a radiation envelope defined by the region inside the cone Z^2=27x^2+48y^2. If the energy density in this...

Suppose I wanted to know the surface integral of a recently whose points are (0,0,0),(0,0,2),(1,1,0),(1,1,2) The integral itself, if the surface is parameterized in terms of u and v, would be in those two variables, a differential element whose sides are du and dv. However, since this...

In "Differential Equations, Dynamical Systems and Introduction to Chaos", the norm of the Jacobian matrix is defined to be: |DF_x| = sup |DF_x (U)|, where U is in R^n and F: R^n -> R^n and the |U| = 1 is under the sup. ...|U| = 1 DF_x (U) is the directional derivative of F in the direction of...

I'm working with a game physics engine that uses Jacobians to resolve contact forces. It's been a few years since my physics and linear algebra classes (where we didn't get to Jacobian matrices), so what I'm reading about Jacobians is fairly overwhelming. Most of what I can find are fairly...

Hello everyone, Does anyone know how I can compute the jacobian matrix numerically in matlab? So, I have the following. A 100x100 image and at each pixel, I have a 2 element gradient vector. What I would like to do is compute the jacobian matrix (wrt to the spatial location), at each pixel...

Homework Statement Find the Jacobian of the transformation: x=\frac{u}{u+v}, y=\frac{v}{u-v} Homework Equations Jacobian = \left|\stackrel{\frac{\partial x}{\partial u}}{\frac{\partial x}{\partial v}} \stackrel{\frac{\partial y}{\partial u}}{\frac{\partial y}{\partial v}}\right|...

Homework Statement The question is in 5 parts all mainly to do with Newton's Method, I've done parts a,b,d and e, but am struggling with the understanding of part c. They have given us the code but unsure as to how to modify my own code to suit the question. Another problem is I don't...

Hello physicists! I'm a comp sci student and I am trying to graphically model a simplified version of the solar system as part of a programming exercise. In order to apply the gravitational forces to the planets, I need to compute the Jacobian matrix as it relates to two particles (planetary...

I'm not sure if this is a stupid question, but I'll go ahead anyway. I understand the math aspect of it, but one thing has me confused. If you have a uv plane, and then write x=x(u,v), y=y(u,v), why is it that no matter what the function transforming the uv plane to the xy plane is, we can...

Hello everyone, So, I read somewhere that the Jacobian determinant of a transformation determines the local volume change. Say I am in 3D space and I have the following relationship: F(x', y', z') = F(x, y, z) + T(x, y, z) The LHS gives the new position and the RHS is the old position + the...

I have the following problem: I'm studying a system of polynomial equations in R^n and I'm looking at the surface which is the solution set of this system. I'm mainly interested in the dimension of this surface at a given point. Now, naively, one would try to compute the Jacobian (of the m...

I'm reading a text on PDEs.. I'm trying to follow some of the argument the author is presenting, but I'm having a bit of difficulty. We start w/ a collection of p functions in n variables (with p <= n). That is to say, we have: u_1, u_2, ..., u_p where u_i : \mathbb{R}^n...

Hi, Assume that I have f(u,v,w,h) du dv dw dh and I need only to change three variables (u, v, w) say to other variables called (s, r, t) and keep h as is it is So my question can I write this as f(u,v,w,h) du dv dw dh = G(r,s,t,h) J(r,s,t) dr ds dt dh where J is jacobian...

What is the relationship between being globally diffeomorphic and the Jacobian of the diffeomorphism? All I can think of is that if the Jacobian at a point is non-zero, then the map is bijective around that point. For example, if: f(x)=x_0+J(x_0)(x-x_0) where J(x0) is the Jacobian...

1. Find the jacobian matrix of the following two equations x'=-16x+3y y' = 18x-19y Homework Equations Here is my attempt I know how a jacobian matrix is derived so is this correct? -16 3 18 -19 The above is meant to be a matrix.

Homework Statement trying to evaluate the double integral from 0 to infinity and 0 to infinity of [(x^2 + y^2)/1 + (x^2-y^2)^2]e^-2xy dxdy using the coordinate transformation u=x^2-y^2 and v=2xy Homework Equations The Attempt at a Solution so i calculated the jacobian...

Homework Statement Y = AX = g(X) Where X,Y are elements of R^n and A is a nxn matrix. What is the Jacobian of this transformation, Jg(x)? Homework Equations N.A. The Attempt at a Solution Well, I know what to do in the non-matrix case. For example... U = g(x,y) V =...

Homework Statement Let D be the image of R = [1; 3] x [1; 4]. under the map T(u; v) = (u^2/v , v^2/u) (a) Compute the Jacobian of T. (b) Compute the area of D. The Attempt at a Solution I'm pretty sure I found the Jacobian (I got -2v/u + 2u/v), but I am confused on the next part...

Homework Statement Here's my question: [PLAIN]http://img140.imageshack.us/img140/1500/89319562.gif The Attempt at a Solution (a) \int^{2 \pi}_0 \int^1_0 [r^2(cos^2\theta + sin^2\theta)]rdr d\theta \int^{2 \pi}_0 \int^1_0 r^3 dr d \theta = \int^{2 \pi}_0 \frac{r^4}{a} |^1_0 d...

The variables u and w are related to x and y by the equations: u=(e^x)*cos(y) and w=(e^-x)sin(y) If I have the Jacobian for δ(u,w)/δ(x,y) How could I manipulate it to find (δx/δw)? With u held constant.

Has anyone ever heard of the "Jacovian" or "Jacobian law/theory the correct spelling for this word might be , "Jacobian" or "Jacovian", or something else, but it is a theory or a law or a rule? Does anyone know what it actually might be called, this is referring to a theory or rule or law that...

I feel so stupid for asking this question, but I want to understand how this integral: \int^{\infty}_{0}d\alpha \int^{\infty}_{0} d\beta \frac{i}{[4 \pi i(\alpha + \beta)]^\frac{D}{2}} e^{[i \frac{\alpha\beta}{\alpha + \beta}p^2 - i(\alpha + \beta)m^2]} can be transformed into this...

When I learn something, especially in calculus, I like to get some intuition on where things come from and why they work, thing is I can't quite fully understand why the jacobian works...I'm not looking for an explanation on how to do it, I already know that, I just want to know why it works, if...

Homework Statement Suppose that P, Q, and R are regions in R2, and suppose T1 : P -> Q and T2 : Q -> R are dierentiable. Use the (multivariable) Chain Rule and det(AB) = det(A)det(B) to show that the Jacobian of the composition T2 o T1 is the product of the Jacobians of T1 and T2...

Homework Statement Suppose R is a plane region bounded by xy=1, xy=3, x^2-y^2=1, x^2-y^2= 4. Use the substitution u=xy, v=x^2-y^2 to evaluate I = \iint\limts_R \, (x^2+y^2) dx\,dy The Attempt at a Solution Using the substitutions given, I find R` = { (u,v) | u for all [1,3] ; v for all [1,4]...

Homework Statement I apologize in advance for my inability to present formal equations here. I'll do my best to be clear with the representation using simple text. "Use the Jacobian Matrix to Prove Laplace's 2D Eq.: (partial^2 u)/(partial x^2) + (partial^2 u)/(partial y^2) = 0"...

Is the following correct, as far as it goes? Suppose I have a vector space V and I'm making a transformation from one coordinate system, "the old system", with coordinates xi, to another, "the new system", with coordinates yi. Where i is an index that runs from 1 to n. Let ei denote the...

Hello everyone, I have a small question about Jacobian and volume changes. So, I have a signal model from an imaging system where the signal intensities are preserved (it's an EPI MRI imaging system). So, basically for volume elements or voxels that are smaller than actually intended, the...

Why is a non zero jacobian the necessary condition for a diffeomorphism? How to prove it?

In differential geometry what does df mean as in \mbox{f} : \mathbb{R}^m \mbox{ to } \mathbb{R}^n Then df is what? the jacobian matrix of partial derivatives?

Homework Statement Use the given transformation to evaluate the given integral. \int\int(x-3y)DA R. where R is the triangular region with vertices (0,0), (2,1) and (1,2) ; x = 2u + v , y = u + 2v Trial : Using the points given I came up with these equations for the...

If f(x,y,z) = xi + yj +zk, prove that Jacobian matrix Df(x,y,z) is the identity matrix of order 3. Because the D operator is linear, D1f(x,y,z) = i, D2f(x,y,z) = k, D3f(x,y,z) = k There is clearly a relationship between this and some sort of identity, but I'm not sure how to state it, and...

Hi, I'm having some problems with the derivation of the Jacobian determinant when used to describe co-ordinate transformations. As I understand it, the Jacobian determinant should relate the areas defined by two vectors in both co-ordinate systems. As the vectors are not necessarily...

I was self-studying the Jacobian and the change in variables when I came upon the following problem: In the integral I = \int_0^\infty \int_0^\infty \frac{x^2 + y^2}{1 + (x^2 - y^2)^2} e^{-2xy} \, dx dy , make the change of variables u = x^2 - y^2 , v = 2xy , and evaluate...

Homework Statement Let f(x,y,z)=(exp(x),cos(y),sin(z)).Compute the Jacobian J(f) of f . In general ,when will the Jacobian J(g) of a function g(x,y,z) be a diagonal matrix ? Homework Equations The Attempt at a Solution I am not quiet sure about this question for J(f) i found...

I've written down the questions I had in the word file , please open and read it so that you can know my problems . Please give me a help! PPT:http://www-astro.physics.ox.ac.uk/~sr/lectures/multiples/Lecture5reallynew.ppt"...

Hi I have a problem. I want to prove a necessary condition in a theorem. I know that a smooth transformation is diffeomorphism around the origin. Can I show that its jacobian is nonsingular at the origin?

calculate the jacobian d(x,y)/d(u,v) of the transformation u=x2+y2 v=x+y for this do i first have to calculate the jacobian d(u,v)/d(x,y) then do 1over the answer? because i would assume the matrix to be det|{(dudx,dudy)(dvdx,dvdy)} but with (u,v) on top i cannot get this

how do we get from this line ||d(x,y,z)/d(p,q,r)||= ||sin(q)cos(r), pcos(q)cos(r), -psin(q)sin(r) end of line 1 line 2 sin(q)sin(r), pcos(q)sin(r), psin(q)cos(r) end of line 2 line 3 cos(q), -psin(q), 0|| to the next line where we take out cos(q) to get cos(q)||pcos(q)cos(r), -sin(q)sin(r)...

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...