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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6 views

Question regarding plotting a Scalar Field that has 3 input variables and 1 output variable

I have a hard time understanding how the equipotential surfaces for a fucntion that has 3 inputs and 1 output is drawn Here ,how is the Contour surface drawn ,is it done by ...
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132 views

Finding the gradient of the restricted function in terms of the gradient of the original function

The following question showed up as part of a proof that I am doing for my research thesis. If we have a differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ and then set $n-d$ coordinates to ...
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1answer
52 views

How to find the directional derivative by definition? [closed]

I have the following function: $$ f(x,y)=3xy^2+e^{xy} $$ I first calculated it in the normal way, using the unit vector, and the resulting value was $\sqrt2$. Then, I tried calculating it by ...
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1answer
57 views

Derivative of matrix inverse with respect to a vector

How can I find the gradient $$\nabla_{u} \left(x^T \left(A \, \mbox{diag}(u)\, A^T \right)^{-1} x \right)$$ where $x \in \mathbb{R}^n$, $u \in \mathbb{R}^d$, are vectors and $A \in \mathbb{R}^{n \...
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1answer
75 views

Vectors of a gradient field [closed]

The scalar field function s is defined by s = (x + y + z)². How do I show that the vectors of the associated gradient field are all parallel, have different lengths and do not always point in the same ...
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1answer
49 views

How to find the gradient $\nabla_W \left( u^T_1 W \left( W E W^T + \lambda I \right)^{-1} W^{T} u_2 \right)$?

I am struggling to find the gradient $$\nabla_W \left( u^T_1 W \left( W E W^T + \lambda I \right)^{-1} W^{T} u_2 \right)$$ where $I \in \mathbb{R}^{n\times n}$ is the identity matrix, $\lambda > 0$,...
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1answer
22 views

Reason for a lower gradient value [closed]

We know that the gradient will be $0$ at maxima or minima. Let's assume we measure the gradient at two points and that at the first point the norm of the gradient is higher than at the second point. ...
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1answer
56 views

Gradient of inner product containing inverse of the sum of two matrices

Given $n \times 1$ vectors $\mathbf{x}$ and $\mathbf{y}$, I need help determining the gradient $$\nabla_{\mathbf{A}} \left( \mathbf{x}^{T} \left( \mathbf{A} + \mathbf{B} \right)^{-1} \mathbf{y} \right)...
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57 views

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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45 views

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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How did they get the final result here?

I am trying to understand the answer of this question. How do you get this? $$\nabla_{\mathrm W}\left(\mbox{tr} \left( \mathrm W^{\top} \mathrm X^{\top} \mathrm X \mathrm W - \mathrm Y^{\top} \mathrm ...
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1answer
69 views

If for any sequence $\vec{r_n}\to\vec{a}$, the sequence $f(\vec{r_n})$ converges to $L$, prove that $\lim_{\vec{r} \to \vec{a}} f(\vec{r}) = L$

Question statement: Let $f : \mathbb{R}^n\to\mathbb{R}$ be a function. If for any sequence $\vec{r_n}\to\vec{a}$, the sequence $f(\vec{r_n})$ converges to $L$, then prove that $\lim_{\vec{r} \to \vec{...
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If $f(x) \leq f(a)$ for all $x$ in an open ball $B(a)$ , then prove that $\nabla f(a) = 0$

Question statement : Assume $f$ is a scalar field function differentiable at each point of an n-ball $B(a)$ . If $f(x) \leq f(a) \ \ \forall \ x $ in an open ball $B(a)$ , then prove that $\nabla f(a)...
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Is there always a solution to Laplace's equation for these boundary conditions?

Consider a region V in 3D space enclosed by a surface S. Let f be some continuous and differentiable function of the three spatial coordinates (a scalar field). Let's say we've specified the gradient ...
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3answers
78 views

Prove that there exists no scalar field which has derivative $> 0$ at a fixed point wrt every direction

The problem statement : Prove that there is no scalar field $f$ such that $\nabla_\vec{y}f(\vec{a})>0$ for a fixed vector $\vec{a}$ and every non-zero vector $\vec{y}$. Here $\nabla_\vec{y}f(\vec{...
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Partial derivative of $x_1^TAB^Tx_2$

Let $A$ and $B$ be orthogonal matrices, and $x_1$, $x_2$ be vectors. What are: $$ \dfrac{\partial}{\partial x_1}\left( x_1^TAB^Tx_2 \right)$$ $$ \dfrac{\partial}{\partial x_2}\left( x_1^TAB^Tx_2 \...
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13 views

Predictable pseudo-random distribution of points in space

I am searching for a method that would allow you to fill a space (2D, 3D) with points based on inputs like fields that would determine point density and other properties. The emphasis is on ...
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25 views

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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1answer
65 views

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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1answer
87 views

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 $A$?
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4answers
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Confused about computing the gradient of least-squares cost

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 ...
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1answer
60 views

Can the gradient exist for a function of $n + 1$ variables?

For a function of $n + 1$ variables $f(x_0, x_1, x_2, ...x_n)$ can a gradient exist? When I asked my professor this during class he said, "no, at most a gradient will exist for a function of ...
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1answer
44 views

How to determine the number of critical points a polynomial scalar field has?

Consider the function $$f(x,y) = x^3 + 3y - y^3 - 3x$$ How would I be able to determine the number of critical points $f(x,y)$ has? I know critical points will exist if $\nabla f(x,y) = 0$ or $f_x(x,y)...
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Is the euclidean norm over a real vector space a scalar field?

One might takes the $\mathbb{R}$-vector space $\mathbb{R}^2$, as I know, it is equivalent to the euclidean plane with a particular point : the origin, so an affine space $(\mathbf{V}_1 =\mathbb{R}^2, \...
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50 views

Which types of matrices conserve the gradient operator?

Let $M \in \Bbb R^{3 \times 3}$. Let $\Omega\subset \mathbb{R}^3$ be a bounded Lipschitz domain and $u :\Omega \to \mathbb R$ be an infinitely-differentiable scalar field. What conditions should ...
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27 views

Can a field be a gradient of a scalar?

I am trying to answer a question in my electromagnetics book (Cheng). I am given a field A with non-zero curl. It is then asked if A can be expressed as a gradient of a scalar. Note that its not ...
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1answer
47 views

How to differentiate 2nd order Taylor expansion of scalar field? [duplicate]

I'm wondering how I can minimize this function with respect to $x$ (not $x_0$). This isn't for homework - I saw them give the answer in the book but they didn't explain how they did it and I'm ...
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1answer
51 views

Gradient of least-squares cost — how to compute it? [duplicate]

I have a system $A x ~ b$ where vector $b$ is not actually in the span of matrix $A$. I want to use a least squares approach to minimize the distance between the two vectors. $$\begin{aligned} \min_x \...
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44 views

Finding the gradient of $𝑓(𝑥,𝑦)=\max\{|𝑥|,|𝑦|\}$ for $|x|=|y|$

We have the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by $$f(x,y) := \max \left\{ |x|, |y| \right\}$$ Let $C := \left\{ (x,y) \mid |x|=|y| \right\}$. Given point $(x,y) \in C$, how can I find ...
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2answers
47 views

Give an $\delta-\epsilon$ proof that the function $f:\mathbb{R}^3\to\mathbb{R}$ defined by $f(x,y,z)=x^2y+2xz^2$ is continuous at $(1,1,1)$

Give an $\delta-\epsilon$ proof that the function $f:\mathbb{R}^3\to\mathbb{R}$ definded by $$f(x,y,z)=x^2y+2xz^2$$ is continuous at $(1,1,1)$. Let $\epsilon<0$ then, $$ \begin{equation} \begin{...
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1answer
66 views

Finding the gradient of $f(x,y)=\max \{|x|,|y| \}$

We have the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by $$f(x,y) := \max \left\{ |x|, |y| \right\}$$ Let $C := \left\{ (x,y) \mid |x|=|y| \right\}$. Given point $(x,y) \notin C$, how can I ...
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24 views

A simple question regarding complex matrix differentiation

I have a complex matrix $\mathbb{A}$ and a complex vector $\mathbb{b}$. How to find the derivative of $$\mathbb{x}^H\mathbb{Ax}$$ and $$\mathbb{b}^H\mathbb{x}$$ with respect to $\mathbb{x}$. Where $\...
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1answer
76 views

Regarding convex functions that attain infimum at infinity

Let function $f: \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable, lower-bounded and convex. Let $\nabla f$ be the gradient of $f$, i.e., $\nabla f = \begin{bmatrix} \frac{\partial f}{\...
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4answers
95 views

Limit $\lim_{(x,y)\to(0,0)}\frac{-16x^3y^3}{(x^4+2y^2)^2}$

Find the limit $$\displaystyle\lim_{(x,y)\to(0,0)}\frac{-16x^3y^3}{(x^4+2y^2)^2}$$ Let $x=r\sin \theta$ and $y=r\cos \theta$. $$ \begin{align} \lim_{(x,y)\to(0,0)}\frac{-16x^3y^3}{(x^4+2y^2)^2}&=\...
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19 views

Gradient of bivariate normal distribution function

Suppose I have a bivariate normal distribution function $$ f(x, y) = \frac{1}{2\pi \sigma_x\sigma_y\sqrt{1 - \rho^2}} \exp\left(-\frac{1}{2(1 - \rho^2)}\left[\left(\frac{x - \mu_x}{\sigma_x}\right)^2 -...
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124 views

Gradient of a quadratic form with respect to a complex vector

How would I go about calculating the derivative with respect to $x$ of $$Q(x) = x^H A x $$ with $A$ a real matrix (not necessarily symmetric) and $x$ a complex valued vector? Here $(\cdot)^H$ denotes ...
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1answer
67 views

Why is the gradient of this determinant, $\det(I - AA^\dagger)$, off by factor of $2$?

I have banged my head on this for a couple of hours and I can't find what's wrong: $$ \begin{align} \frac{\partial}{\partial A_{pq}}\det(\mathbb{1} - AA^\dagger) &= \det(\mathbb{1} - AA^\dagger)\...
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2answers
78 views

Partial derivatives of $|Ax - b|^2$? [duplicate]

I'm trying to work out the partial derivatives of a function $L$ in terms of $x_i$: $$ A \in \mathbb{R}^{m x n} \quad b \in \mathbb{R}^m \quad x \in \mathbb{R}^n $$ $$\begin{aligned} L(x) &= \left\...
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25 views

Explanation for dimension mismatch of Hessian of $f(x) = \log(1+x^TQx)$ where $Q=Q^T$ and $Q$ is positive-definite

According to this link, we have that given $f(x) = \log(1+x^TQx)$, the Hessian matrix is given by $$\nabla^2f(x)=\frac{Q^T+Q}{1+x^TQx}- \left( \frac{(Q+Q^T)x}{1+x^TQx}\right)^T \left( \frac{(Q+Q^T)x}{...
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1answer
43 views

Strongest increase of a gradient $f(x,y)$ at the position $x_0$

I am trying to figure out how to interpret the gradient of my scalar field at the position $x_0$. The gradient of the function $$f: \mathbb{R}^n \supset \ U \longrightarrow \mathbb {R}$$ at the point $...
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1answer
67 views

Derivative of trace involving Hadamard product

Let us assume that $A, S\in\mathbb{R}^{n\times n}$, $U\in\mathbb{R}^{n\times k}$, and $V\in\mathbb{R}^{n\times k}$. I am trying to differentiate the following expression: $$\Phi(U,V)=\mathrm{trace}\...
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1answer
41 views

Features of a given scalar field [closed]

I have to create a visualization of a scalar field given by the formular: $$f(x,y) = x^3 - 3xy^2$$ I have to represent some features of this scalar field. I plotted the following scalar fieldbut can´t ...
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2answers
83 views

Differentiate vector transpose using rules [duplicate]

I am referring to Tom Minka's Old and New Matrix Algebra Useful for Statistics. I don't have the book by Magnus & Neudecker so I can't refer to the details of the theory. Regarding rules (6): $d(...
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2answers
37 views

Gradient of function that can be expressed as another function

Given unknown constants $a$ and $b$, let $$f(x,y) := \frac{a}{2} x^2+ \frac{b}{2} y^2$$ meaning that $$\nabla f(x,y) = \begin{bmatrix} ax\\ by\end{bmatrix}$$ Then I have $h(k) = f((1-ka)x_0, (1-kb)y_0)...
3
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1answer
128 views

Get wrong answer on $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}$ when using graph

I can use the product rule to obtain $\frac{\partial \mathbf{x}^{\top} \mathbf{A} \mathbf{x}}{\partial \mathbf{x}} = \mathbf{x}^{\top} \frac{\partial \mathbf{A} \mathbf{x}}{\partial \mathbf{x}}+(\...
0
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1answer
107 views

If the scalar product of two vectors is equal to the magnitude of their vector product, find the angle between them. [closed]

I found some similar questions but none could satisfy me. I am not given any other conditions except those mentioned above. Please help me with this question. I am mainly confused on the formula for ...
1
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2answers
53 views

Gradient direction / descent

it was a while ago I read multivariable calculus so I need to refresh certain results. Given $ f:R^n\to R $, at a local stationary point $ x $ the gradient is $ \nabla f(x) = 0 $. However, given the ...
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0answers
24 views

Integrating over a conservative field

If $\mathbf{\underline{F}}$ is a conservative vector field and $\phi$ is a scalar field defined as: $\phi(\mathbf{\underline{x}}) = \int_{\mathbf{\underline{0}}}^{\mathbf{\underline{x}}} \mathbf{\...
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58 views

Showing $∇$ x $\left(f^2∇f\right)=0$

I am trying to show that $∇$ x $\left(f^2∇f\right)=0$ (the zero vector) I know that from the expansion we get $∇f^2$ x $∇f\:+\:f^2∇$ x $∇f$ From this we can say that $f^2∇$ x $∇f$ = 0 as the curve of ...

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