# Questions tagged [scalar-fields]

A scalar field is a function of the type $X\to \Bbb R$, where $X$ may be an open set in $\mathbb R^n$ or more generally a smooth manifold.

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I have an ensemble of Gaussian random fields $X(s)$ over coordinates $s$. The pdf of an individual realisation of the field taking a value $x(s)$ at coordinate $s$ is given by $$f_x(x(s))=\frac{1}{(2\... • 33 1 vote 1 answer 29 views ### Calculating gradient ascent vector from a scalar field For a computer program, given an nxn grid of scalar values (which can be viewed as a height field), I need to calculate the ... • 119 1 vote 1 answer 32 views ### Hessian of quadratic objective function I have the quadratic function$$f(\boldsymbol{x}) = \frac{3}{2} \left (x_{1}^{2}+x_{2}^{2} \right) + (1+a) x_{1} x_{2} - \left(x_{1} + x_{2} \right) + b$$where a, b \in \Bbb R are unknown ... • 121 2 votes 1 answer 79 views ### Directional derivative definition versus gradient Given the following scalar field$$f(x,y) = \begin{cases} \frac{y^3}{x^2+y^2} & (x,y)\ne(0,0) \\ 0 & (x,y)=(0,0) \end{cases}$$find its directional derivative ... 1 vote 2 answers 63 views ### Time derivative of the function  t \mapsto \frac{1}{2}\dot{f}(t)^T A (f)\dot{f}(t)  I have been trying to differentiate the following function with respect to time$$ t \mapsto \frac{1}{2}\dot{f}(t)^T A (f)\dot{f}(t) $$but I am struggling with the chain rule for matrix calculus. ... 1 vote 0 answers 62 views ### Minimizing \|A−XB\|^2_F Find a closed form solution for$$\underset{X}{\operatorname{argmin}} \|A−XB\|^2_F$$where \| \cdot \|_F denotes the Frobenius norm. I have found this thread. Not sure if it helps here though. Any ... 1 vote 1 answer 51 views ### Derivative of diagonal matrix \mbox{diag}(X 1_n)  with respect to X For a symmetric matrix X \in \Bbb R^{n\times n}, let$$f(X) := u^\top \mbox{diag}(X 1_n) v$$What is the derivative of f with respect to X? X 1_n is the row-wise summation to generate a vector ... • 211 1 vote 1 answer 63 views ### Differential of quadratic form I'm having trouble understanding this: if \phi(x)=x^{T}Ax, where A is a matrix of constraints. Then, the differential of \phi: \mathrm{d} \phi=(\mathrm{d} x)^{T} A x+x^{T} A \mathrm{~d} x=x^{T} ... • 13 0 votes 1 answer 43 views ### Gradient of matrix function using the trace For the function \label{eq:sparsecost} \mathcal{C}\left(\mathbf{B}, \mathbf{A}\right) = \frac{1}{K} \sum_{k=1}^K \left ( \sum_j \left[ {S}_{kj} - \sum_i {A}_{ki} \: {B}_{ij} \right]^2 ... 1 vote 1 answer 49 views ### Correct definition of derivatives of higher dimensions - Derivative of quadratic form - From the perspective of notation I am studying optimal control (for an introductory course) and we have to know the derivatives of quadratic forms. Out teacher said that the following are true:$$\frac{\partial Ax}{\partial x} = A^T $... 0 votes 3 answers 60 views ### Gradient of$\mbox{trace}(Axx^TB)$I am trying to find the gradient $$\nabla \mbox{trace}(Axx^TB)$$ where both$A$and$B$are$n \times n$matrices, and$x$is an$n$-length column vector I'm not exactly sure how to approach this ... 0 votes 2 answers 71 views ### Can the field (underlying a vector space) always be considered a 1-dimensional vector space? For example. We know that a linear transformation is exactly the unique homomorphism between vector spaces, therefore applying a linear transformation to a vector always leads to a vector. At the ... 1 vote 0 answers 53 views ### Correlation of the convolution between a uniformly distributed random noise and a Gaussian function Given a random scalar field$u(x,y)$for$x,y \in D \subset \mathbb{R}^2$where$D$is a rectangular box centered at$(0,0)$, the value of$u(x,y)$at an arbitrary point$(x,y)$is uniformly ... • 143 -1 votes 1 answer 54 views ### Can a unit vector be treated as a vector field? Say you have the kth component of a vector field: $$\phi \hat{k}$$ Can we treat this as a scalar field$\phi$multiplied by the vector field$\hat{k}$? EDIT: So, if you wanted to find the curl of$\...
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I'm struggling with this very simple gradient question. Where... Now I've already been told the answer is Ax + b, but I can't work out how to get to the first term. Intuitively it looks like a ...
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### Gradient of linear scalar field with respect to matrix

I am following the book Mathematics for Machine Learning to study the math necessary to understand machine learning papers. On page 158, the authors list these results about gradients of scalar fields ...
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### Proof for the Laplacian of scalar fields using index notation [closed]

For scalar fields Φ and Ψ, the Laplacian is defined by $$∇^2(ΦΨ)=(∇^2Φ)Ψ+2∇Φ\cdot∇Ψ+Φ∇^2Ψ$$ where $∇$ is the usual del operator and $∇^2$ is the Laplacian. How can I prove this relation? I tried the ...
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### Clarifying Eq. 101 in The Matrix Cookbook

Eq. 101 of The Matrix Cookbook claims that $$\frac{\partial \operatorname {Tr}(AXB)}{\partial X} = A^\top B^\top$$ The point that is unclear to me is why the dimensions in the product fit? For example,...
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### Can I swap the positions in this equation?

I am looking at this question. I wonder if I have: $$\left\|Y-XW\right\|_{\text{F}}^2$$ Does taking derivative w.r.t $W$ yields this? $$-2X^T(Y-XW)$$
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### Is there a bundle concept that includes tensors and spinors as special cases?

A scalar field is a map from the base space to the field of interest but it is equivalently a section of a (0,0)-tensor bundle. Similarly, a vector bundle is just a section of a (1,0)-tensor bundle. ...
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### Matrix differentiation, $x^T A x$ w.r.t $A$ [duplicate]

In Gaussian Mixture Models, in order to derive the M-step for covariance matrix, I need this result. I have poor knowledge of matrix calculus and result does not exist in https://en.wikipedia.org/wiki/...
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### Finding Potential for specific Vector Field, help?

I've been trying to think of a potential to derive for the vector field $$\vec{F}(r,\theta,z) =-mg\sin(\theta) \hat{e}_{\theta} .$$ I'm assuming this is a force as in physics so the potential is ...
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### What is the gradient of $x^T A\, x$ with respect to the matrix $A$? [duplicate]
I have seen many times that the gradient of $x^TA\,x$ with respect to $x$ is $2A\,x$. But how do you find its gradient with respect to $A$?
Given matrix $A \in \mathbb R^{m \times n}$ and vector $y \in \mathbb R^m$, I want to take the gradient of the following scalar field with respect to $x\in \mathbb R^n$. x \mapsto \big((Ax - y)^T(Ax ...