# Questions tagged [jacobian]

In multivariable calculus, the jacobian matrix of a smooth map at a given point is the matrix of its partial derivatives evaluated at this point.

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### The log Jacobian of transforming a normal CDF to Dirichlet Random Variable transformation?

I have random variables $\mu$ and $\sigma$ that I have transformed and I am interested in finding their joint distribution given the following information I have. Particularly, I need help finding the ...
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### Why is the Jacobian of this random vector transformation the determinant of the fixed matrix?

Here's a link: https://www.probabilitycourse.com/chapter6/6_1_5_random_vectors.php. My question concerns example 6.15 Here is the stated problem: Let $\mathbf{X}$ be an $n$-dimensional random vector. ...
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### The Jacobian of a $f':\Bbb{R}^n\to\cal{L}(\Bbb{R}^n;\Bbb{R}^n)$

I was given a function $f:\Bbb{R}^n\to\Bbb{R}^n$ that is $\cal{C}^{\infty}$ and was asked to calculate the Jacobian matrix of its derivative $f':\Bbb{R}^n\to\cal{L}(\Bbb{R}^n,\Bbb{R}^n)$. I know from ...
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### Jacobian with zero eigenvalues - Fixed points

I'd like to ask, if possible, for some good and not excessively long material about classification of fixed points in which the Jacobian has zero eigenvalues. Many thanks in advance!
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### Eigenvalues of Jacobian of Mandelbulb “triplex” power formula

I'm trying to find a lower bound for the distance estimate of the Mandelbulb fractal, or at least justify why using the scalar-derivative for distance estimation is so effective. The Mandelbulb ...
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I was studying a literature where in an equation the term $\nabla p_i$ is given and further it is given that $\nabla$ is a column wise gradient and $p_i = p_i(x,t)$, what exactly meaning of column-...
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### What does this equation mean? Shouldn't the solution be a 1x3 matrix instead of a value?

If we have a function then what is equal to? The solutions are all single values, but if D(1,2,3) is the jacobian matrix on the point (1,2,3) and f(3,2,1) the function on that point, it should give ...
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### How Is the Jacobian squared ( truncated Hessian) approximately equal to the Hessian [closed]

I'm currently work on second order optimization. And I'm stuck at quass Newton method. I'm are trying to approximate the Hessian without actually computing the Hessian . Please tell how can the ...
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### Jacobi preconditioned CG method - diagonal

Would you say that in the Jacobi preconditioned CG method of the diagonal of Q^-1 is always well defined? In the quadratic case.
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### Jacobian and Inverse Jacobian

I am new to Jacobians and still trying to understand how they work. My understanding so far is as follows: Suppose I have a function $f$ expressed as: $$f(b_1(a_1,a_2),b_2(a_1,a_2),b_3(a_1,a_2))$$ ...
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### How to find slope of locus?

Given a system of DEs: $$\begin{cases} \dot{x} = g_1(x,y) \\ \dot{y} = g_2(x,y), \end{cases}$$ where $g_1,g_2 \in C^1$ How to show that the slope of $g_1 = 0$ in the neighborhood of the steady state ...
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### Stability of these fixed points

Say we have the set of nonlinear equations, where $\alpha>0$: $$\begin{matrix} \frac{dx}{dt}=x[1-\alpha x-y]\\ \frac{dy}{dt}=y[1-x-\alpha y] \end{matrix}$$ I have determined that the fixed ...
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The surface integral over surface $S$ (which is given by $z=f(x,y)$, where $(x,y)$ is a point from the region $D$ in the $xy$-plane) is: $$\iint\limits_{S} g(x,y,z)\ dS = \iint\limits_{D} g(x,y,f(x,... 1answer 34 views ### How do I handle this probability density function with a Jacobian? "Suppose X and Y are independent random variables, each exponentially distributed with parameter \lambda. Determine the probability density function for Z=\frac{X}{Y}." Here is what I have so far:... 1answer 43 views ### Generalization of Gradient Using Jacobian, Hessian, Wronskian, and Laplacian? I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of ... 2answers 196 views ### What is the difference between the Jacobian, Hessian and the Gradient? I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of ... 2answers 40 views ### Is a constant Jacobian an equivalent condition to linearity? [closed] And furthermore, does a constant, non-zero Jacobian imply that the function is invertible? 1answer 43 views ### How can I show \mathbf{e}_0\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3=\sqrt{|g|}\gamma_0\gamma_1\gamma_2\gamma_3 Let me try to work it out with Cl_2(\mathbb{R}) with basis elements \hat{x}, \hat{y} such that \hat{x}^2=1, \hat{y}^2=1, \hat{x}\hat{y}+\hat{y}\hat{x}=0. I define a non-orthonomal basis as ... 0answers 28 views ### Transformation of Lie Algebra By Modifying Generating Vector Fields Let U_1,\dots,U_k be C^1-vector fields on \mathbb{R}^n and let f:\mathbb{R}^n\to (0,\infty) be a C^1-function. Let Lie(U_i) denote the Lie algebra generated by \{U_i\}_i=1^k. When is ... 0answers 24 views ### Jacobian Matrix and Exponential Map? Let f:\mathbb{R}^n\to\mathbb{R}^n be a diffeomorphism. Can f always be represented as$$ f(x)=\exp(F(x))x,  where $F:\mathbb{R}^n\to Mat_{n\times n}$? Intuition/Direction Somehow it seems ...
In exercise 17 of chapter 9, Rudin lets a function from $\mathbb{R}^2$ to itself be defined by $f_1(x,y)=e^x\cos(y)$, $f_2(x,y)=e^x\sin(y)$. Rather than a solution to the exercise, I am interested in ...