Jacobian Definition and 163 Threads

I am computing matrix elements of a two body quantum-mechanical potential, which take the form V_{k l m n} = \int d^3 r_1 d^3 r_2 e^{-i k \cdot r_1} e^{-i l \cdot r_2} V( | r_1-r_2 | ) e^{i m \cdot r_1} e^{i n \cdot r_2} To do this integral, I make the change of coordinates...

Hi, I have two numerical matrices, A is 150*1 matrix (A=rand(150,1)) and B is a 1*5 matrix (B=rand(1,7)), and I need to have the jacobian of A with respect to B, that should be a 150*7 matrix, anyone help is appreciated. Mathias

Dear all, I was revising on a bit of tensor calculus, when I stumbled upon this: $$\delta^i_j = \frac{\partial y^i}{\partial x^\alpha} \frac{\partial x^\alpha}{\partial y^j}$$ And the next statement reads, "this expression yields: $$|\frac{\partial y^i}{\partial x^j}|...

x=t^2-s^2, y=ts,u=x,v=-y a) compute derivative matrices \vec{D}f(x,y) = \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right] \vec{D}f(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right] b) express (u,v) in terms of (t,s) f(u(x,y),v(x,y) = (t^2-s^2,-(ts)) c) Evaluate \vec{D}(u,v)...

I just wanted to verify which order you put the rows of the Jacobian. If your initial variables are (x,y) and change to new variables u(x,y)=f(x,y) and v(x,y)=g(x,y), then you'll get a Jacobian. If this Jacobian is negative, would you change your definition to u(x,y)=g(x,y) and v(x,y)=f(x,y)...

Umm what just happened? I understand as far as u=x+y and v = y/x and when he does the 2d curl. What I don't get is the step thereafter when he flips it. How does he know to flip it? Further, when he flips it wouldn't that make the dvdu inside the integral cancel and hence leave him with dxdy?

Homework Statement Homework Equations The Attempt at a Solution What I don't understand is how I'm supposed to find those limits for U and V. That's not at all what I'm getting. I've tried solving for the max and min x,y coordinates from the given graphs but that doesn't yield...

Hello, I'm trying to make my own physics engine. I've already made one with momentum and an very easy collision solver, but it was creepy and full of bugs because I solved the contacts one after another. Then I read about the difference between iterative and single pass algotrithms and I...

Suppose we do a constant Jacobian transformation (which is not Lorentz) of a SR (inertial) frame, by using four linear change of variables equations. This defines an apparent field with a constant metric (which is not the SR metric) in which there is relative acceleration of separation. From...

In somewhere in wikipedia, I found a "shortcut" for compute the jacobian, the formula is: \frac{\partial(q_1 , q_2 , q_3)}{\partial (x, y, z)} = h_1 h_2 h_3 where q represents the coordinate of other system and h its factor of scale. I know that this relationship is true. What I'd like of...

Hello! :o Having the transformation t=τ+p, I want to calculate the jacobian $\frac{J(t,τ)}{J(τ,p)}$. Isn't it $$ \frac{J(t,τ)}{J(τ,p)}=\begin{vmatrix} \frac{ \vartheta t}{\vartheta τ}& \frac{\vartheta t}{\vartheta p} \\ \frac{\vartheta τ}{\vartheta τ} & \frac{\varthetaτ }{\vartheta p}...

Hi, Started to learn about Jacobians recently and found something I do not understand. Say there is a vector field F(r, phi, theta), and I want to find the flux across the surface of a sphere. eg: ∫∫F⋅dA Do I need to use the Jacobian if the function is already in spherical...

Homework Statement Let F: x^2 + y^2 - z^2 + 2xy - 1 = 0 and G: x^3 + y^3 - 5y - 4 = 0. Calculate dz/dx. Note: This is NOT the partial derivative ∂z/∂x. I do not need help in taking the derivative of many polynomials. What I need help in is setting up a Jacobian determinant to evaluate this...

Homework Statement Find surface inside four boundary curves: xy = 4 , xy=8 , y=5x , y=15x using the transformation: u=xy , v=\frac{y}{x} Homework Equations I'm getting the new bounds to be: 4 < u < 8 , -15 < v < -5 OR 5 < v < 15 Jacobian is \frac{1}{2v}The Attempt at a...

##\phi:\mathbb R^4\to\mathbb R^4## is a smooth function such that ##J_\phi(x)^T\eta J_\phi(x)=\eta##, where ##J_\phi(x)## is the Jacobian matrix of ##\phi## at x, and ##\eta## is defined by $$\eta=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}.$$ I...

Homework Statement Hello:) My problem is as follows: Determine the following commutators: [px2,x],[pxx2],[px2,x2],[]. The calculation can be done in two ways, either by inserting a test function, and using the explicit expressions for the operators, or by utilizing Jacobi identity and using...

I'm working on an inverse kinematics problem (I make video games), and I'm reaching a bit beyond my education. Right now, I've got an algorithm that solves the basic IK equation for a chain of rigid bodies connected by joints by approximately inverting ##J\Delta\theta = e##. Where ##\theta##...

Hi, When I do the following transformation: $$ X_1=x_1+x_2 \\ X_2=x_2 $$ It turns out that the Jacobian ##\partial (X_1,X_2)/\partial (x_1,x_2)## is 1. But we have: $$ dx_1dx_1+dx_1dx_2=d(x_1+x_2)dx_2=dX_1dX_2=|\partial (X_1,X_2)/\partial (x_1,x_2)|dx_1dx_2=dx_1dx_2 $$ So we...

So here is my problem: I'm very bad when using coordinates others than cartesian ones, and I know taking Mechanics of Lagrange and Hamilton and I fin difficult to find the velocities expressions in curvilinear coordinates. So here is my question: is there anyway to relate the scale factors...

Hello, I was starting with an equation using hooke's law and with a damping factor appended to it. The equation is to calculate the x,y, and z components of the resulting force, as a vector. The hooke's law portion uses the current position in calculating the force (x, y, and z position...

In what kind of math course would one learn the proof of the theorem that introduces the Jacobian to computing multiple integrals under various transformations? My calculus textbook has this theorem, and uses it to derive the triple integral formulas for cylindrical and spherical coordinates...

Homework Statement Change of coordinates from rectangular (x,y) to polar (r,θ). Not sure what's wrong with my working.. Homework Equations The Attempt at a Solution

Not sure if this is where I should put this but currently I am taking math for econ and we are on special determinants (jacobian, Hessian, Bordered Hessian, some Leontiff) So I have this problem in my notes that I am basically basing my exam studying around since the book isn't the best. It...

Hi, Homework Statement I was hoping someone could please explain the order of variables in a Jacobian. I mean, once the dependent and independent variables have been identified, how should the Jacobian be formulated. For instance, supposing I have two implicit functions F(x,y,u,v) and...

Hello, let's suppose I have two functions \phi:U\rightarrow V, and T:V\rightarrow V that are both diffeomorphisms having inverse. Furthermore T is linear. I consider the function f(u) = (\phi^{-1}\circ T \circ \phi)(u), where \circ is the composition of functions. Since T is linear, we...

Hello, it is true that linear transformations have constant Jacobian determinant. Is the converse true? That is, if a transformation has constant Jacobian determinant, then is it necessarily linear?

Consider: \int d\phi e^{iS[\phi]}=\int d\phi' J e^{iS'[\phi']} where J is the Jacobian. If the transformation of variables to phi' is a symmetry of the action [i.e., S'=S], then this becomes: \int d\phi e^{iS[\phi]}=\int d\phi' J e^{iS[\phi']} But doesn't this imply that the Jacobian has...

I am dealing with a random variable which is a transformation of another random variable of the form: Y:=aX^b+c The pdf of the random variable X is known and for the sake of example let it be exponential distribution or any other distribution with known and commonly available quantile...

I am trying to understand the role of the Jacobian in the Implicit Function Theorem. However, I have had a hard time finding any discussions that use the Jacobian and are accessible for my level. http://mathworld.wolfram.com/ImplicitFunctionTheorem.html has been the best thing I have found...

Why is it that if you have: U=g_1 (x, y), \quad V = g_2 (x,y) X = h_1 (u,v), \quad Y = h_2 (u,v) Then: f_{U,V} (u,v) du dv = f_{X,Y} (h_1(u,v), h_2 (u,v)) \left|J(h_1(u,v),h_2(u,v))\right|^{-1} dxdy While when doing variable transformations in calculus, you have: du dv =...

Before I ask my question, I'll lead up to it through an example. Just for reference, I have only taken up to Calc 3 and haven't taken Vector Calc. Let's look at this definite integral: ∫∫cos(x^2+y^2)dxdy The bounds on the outer integral is from 0 to 1 while the bounds on the inner integral...

Homework Statement I need to compute the following using the Jacobian: \int\int_D \frac{x-y}{x+y} dxdy Where D = \left\{(x,y):x\geq 0, y\geq 0, x+y \leq 1\right\} Homework Equations The Attempt at a Solution I've made the transformation: s=x+y \qquad t = x-y My problem is finding the...

Hi, I need some help understanding the solution to a problem. Equations: x = r.cos(θ) y = r.sin(θ) r = x2 + y2 theta = arctan(y/x)Question: Determine the Jacobian Matrix for (x,y)T and for (r, θ)T SOLUTION: I understand and can compute by myself the Jacobian for (x,y)T, but the solution to...

Homework Statement what routine algebra are they talking about. I don't see how they got x = u/3 - v/3 or y = 2u/3 +v/3

Dear Forum, I have been given a DT fusion spectrum where neutrons are produced in the following reactions, DT = 14.1 MeV DD = 2.45 MeV TT = 0-9.8 MeV (three body spectrum) The spectrum is given in the form dN/dE. I have converted this spectrum into the dE/dE by multipling the...

Hi: I`m new here, can someone tell me which is the difference between the transition matrix and the Jacobian?, I did some exercises of the both topics, but How is it related? Thanks for the attention Sorry for my English writing, but English is not my native language.

What exactly does it mean when the determinant of a Jacobian matrix vanishes? Does that imply that the coordinate transformation is not a good one? How do you know if you coordinate transformation is a good one or a bad one?

Homework Statement Show that T(u,v) = (u2 - v2, 2uv) maps to the triangle = {(u,v): 0 ≤ v ≤ u ≤ 4} to the domain D bounded by x=0, y=0, and y2 = 1024 - 64x. Use T to evaluate ∬D sqrt(x2+y2) dxdy Homework Equations The Attempt at a Solution x=u2-v2 y=2uv Jacobian= 4u2+4v2 dudv I guess the...

Homework Statement For the transformation, draw the lattice lines, calculate the derivative, and calculate the Jacobian. x=rcos\theta y=rsin\theta The Attempt at a Solution I drew the lattice lines correctly. What I am confused about is the derivative. Since x and y are both...

So u-substitution is used to make an effective change of variables in one-dimensional calculus. Jacobian determinants are used to make a change of variables in two or higher-dimensional calculus. Can a u-substitution be thought of as a one-dimensional Jacobian determinant? And on an entirely...

Hi, I have a function of the form: function r=d(f,j) if (j==1) r=3*f(1)+f(3)*f(2)-3; elseif(j==2) r=f(1)+2*f(2)-4; else r=f(1)*f(2)*f(3)-1; end and I would like to be able to numerically compute the Jacobian for such a function. I have a method of computing the Jacobian...

Homework Statement Evaluate the surface integral. ∫∫S x^2*z^2 dS S is the part of the cone z^2 = x^2 + y^2 that lies between the planes z = 1 and z = 3. Homework Equations \int \int _{S}F dS = \int \int _D F(r(u,v))|r_u\times r_v|dA x=rcos(\theta) y=rsin(\theta) The Attempt...

Homework Statement Evaulate the integral making an appropriate change of variables. \int\int_R(x+y)e^{x^2-y^2}dA where R is the parallelogram enclosed by the lines x-2y=0, x-2y=4, 3x-y=1, 3x-y=8 . Homework Equations The Attempt at a Solution I'm not sure what change of variables I should...

Greetings all, I hope someone out there in the vast hinterland of the internet can help. I'm trying to calculate the lyapunov exponent for a system of differential equations. Now I can do this just fine for a system involving only first order derivatives such the Lorenz system, however, and...

There are 2 parameters in the Gamma distribution, alpha and beta. If sample 500 of the Gamma random variable, there unbiased mean and variance can be estimated by the sample moments. If it is also interested to estimate the variance and covariance of the parameters, alpha and beta; Jacobian...

Can someone direct me to a good deep exposition of Jacobians and Hessians? I am especially looking for stuff that pertains to their being generalizations of derivatives of vector and scalar functions as well as div, grad, curl. Book sources or web links are appreciated.

Hello! I recently tried to prove following theorem: Let \phi:B\to\mathbb{R}^2 be a diffeomorphism (regular, injective mapping). Then \int_{\phi(B)}f(\mathbf{x})\,\mathrm{d}x=\int_{B}f(\phi(\mathbf{t}))\left|{\mathrm{det}}\mathbf{J}_{\phi}\right|\mathrm{d}t With following I can't proof...

Hi guys, let's say I have a transformation T from (p,q) to (u,v). The inverse transformation would be T^{-1} from (u,v) to (p,q) Now, J(T) = u_{p}v_{q} - u_{q}v_{p}. On the other hand, J(T^{-1})= p_{u}q_v - p_{v}q_{u}. But |J(T)J(T^{-1})| = 0 and not equal to 1. I know it's supposed to be 1...

Homework Statement Consider the scalar field Φ(x, y, z) given by Φ(x, y, z) = x^2y^2z^2 (a) Write down the vector field A(x, y, z) defined by A(x, y, z) = ∇Φ(x, y, z) (b) Write down the scalar field Q(x, y, z) defined by Q(x, y, z) = ∇ . A(x, y, z) (c) Find the Jacobian for the...

I get the idea of Jacobian matrices. I think. Working through different examples, I don't have any problems. For example, f1 = x^2 + y^2 f2 = 3x + 4y would result in [2x 2y] [3 4] Similarly, by my understanding, something like x^2 + y^2 3y + 4x would result in [2x...