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.
252
questions
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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 ...
5
votes
3answers
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 ...
0
votes
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 ...
0
votes
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 \...
0
votes
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 ...
0
votes
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, $\lambda > 0$,...
-3
votes
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. ...
0
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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
$$\nabla_{\mathbf{A}} \left( \mathbf{x}^{T} \left( \mathbf{A} + \mathbf{B} \right)^{-1} \mathbf{y} \right)...
0
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0answers
57 views
Clarifying Eq. 101 in The Matrix Cookbook
Eq. 101 of The Matrix Cookbook claims that
$$\frac{\partial \operatorname {Tr}(AXB)}{\partial X} = A^\top B^\top$$
The point that is unclear to me is why the dimensions in the product fit? For example,...
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0answers
45 views
Can I swap the positions in this equation?
I am looking at this question. I wonder if I have:
$$\left\|Y-XW\right\|_{\text{F}}^2$$
Does taking derivative w.r.t $W$ yields this?
$$-2X^T(Y-XW)$$
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3answers
45 views
How did they get the final result here?
I am trying to understand the answer of this question. How do you get this?
$$\nabla_{\mathrm W}\left(\mbox{tr} \left( \mathrm W^{\top} \mathrm X^{\top} \mathrm X \mathrm W - \mathrm Y^{\top} \mathrm ...
0
votes
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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2answers
109 views
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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0answers
27 views
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 ...
2
votes
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{...
1
vote
2answers
47 views
Partial derivative of $x_1^TAB^Tx_2$
Let $A$ and $B$ be orthogonal matrices, and $x_1$, $x_2$ be vectors. What are:
$$ \dfrac{\partial}{\partial x_1}\left( x_1^TAB^Tx_2 \right)$$
$$ \dfrac{\partial}{\partial x_2}\left( x_1^TAB^Tx_2 \...
0
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0answers
13 views
Predictable pseudo-random distribution of points in space
I am searching for a method that would allow you to fill a space (2D, 3D) with points based on inputs like fields that would determine point density and other properties. The emphasis is on ...
0
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0answers
25 views
Is there a bundle concept that includes tensors and spinors as special cases?
A scalar field is a map from the base space to the field of interest but it is equivalently a section of a (0,0)-tensor bundle. Similarly, a vector bundle is just a section of a (1,0)-tensor bundle. ...
0
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1answer
65 views
Matrix differentiation, $x^T A x$ w.r.t $A$ [duplicate]
In Gaussian Mixture Models, in order to derive the M-step for covariance matrix, I need this result. I have poor knowledge of matrix calculus and result does not exist in https://en.wikipedia.org/wiki/...
0
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0answers
16 views
Finding Potential for specific Vector Field, help?
I've been trying to think of a potential to derive for the vector field
$$ \vec{F}(r,\theta,z) =-mg\sin(\theta) \hat{e}_{\theta} .$$
I'm assuming this is a force as in physics so the potential is ...
1
vote
1answer
87 views
What is the gradient of $x^T A\, x$ with respect to the matrix $A$? [duplicate]
I have seen many times that the gradient of $x^TA\,x$ with respect to $x$ is $2A\,x$. But how do you find its gradient with respect to $A$?
3
votes
4answers
96 views
Confused about computing the gradient of least-squares cost
Given matrix $A \in \mathbb R^{m \times n}$ and vector $y \in \mathbb R^m$, I want to take the gradient of the following scalar field with respect to $x\in \mathbb R^n$.
$$x \mapsto \big((Ax - y)^T(Ax ...
1
vote
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 ...
0
votes
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)...
0
votes
0answers
32 views
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, \...
0
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0answers
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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0answers
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 ...
0
votes
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 ...
0
votes
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.
$$\begin{aligned} \min_x \...
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0answers
44 views
Finding the gradient of $𝑓(𝑥,𝑦)=\max\{|𝑥|,|𝑦|\}$ for $|x|=|y|$
We have the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by
$$f(x,y) := \max \left\{ |x|, |y| \right\}$$
Let $C := \left\{ (x,y) \mid |x|=|y| \right\}$. Given point $(x,y) \in C$, how can I find ...
0
votes
2answers
47 views
Give an $\delta-\epsilon$ proof that the function $f:\mathbb{R}^3\to\mathbb{R}$ defined by $f(x,y,z)=x^2y+2xz^2$ is continuous at $(1,1,1)$
Give an $\delta-\epsilon$ proof that the function $f:\mathbb{R}^3\to\mathbb{R}$ definded by $$f(x,y,z)=x^2y+2xz^2$$ is continuous at $(1,1,1)$.
Let $\epsilon<0$ then,
$$
\begin{equation}
\begin{...
0
votes
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 ...
0
votes
0answers
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 $\...
6
votes
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}{\...
0
votes
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 $y=r\cos \theta$.
$$
\begin{align}
\lim_{(x,y)\to(0,0)}\frac{-16x^3y^3}{(x^4+2y^2)^2}&=\...
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0answers
19 views
Gradient of bivariate normal distribution function
Suppose I have a bivariate normal distribution function
$$
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 -...
1
vote
0answers
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 ...
0
votes
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:
$$
\begin{align}
\frac{\partial}{\partial A_{pq}}\det(\mathbb{1} - AA^\dagger) &= \det(\mathbb{1} - AA^\dagger)\...
1
vote
2answers
78 views
Partial derivatives of $|Ax - b|^2$? [duplicate]
I'm trying to work out the partial derivatives of a function $L$ in terms of $x_i$:
$$ 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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votes
0answers
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 $$\nabla^2f(x)=\frac{Q^T+Q}{1+x^TQx}- \left( \frac{(Q+Q^T)x}{1+x^TQx}\right)^T \left( \frac{(Q+Q^T)x}{...
1
vote
1answer
43 views
Strongest increase of a gradient $f(x,y)$ at the position $x_0$
I am trying to figure out how to interpret the gradient of my scalar field at the position $x_0$.
The gradient of the function $$f: \mathbb{R}^n \supset \ U \longrightarrow \mathbb {R}$$ at the point $...
2
votes
1answer
67 views
Derivative of trace involving Hadamard product
Let us assume that $A, S\in\mathbb{R}^{n\times n}$, $U\in\mathbb{R}^{n\times k}$, and $V\in\mathbb{R}^{n\times k}$. I am trying to differentiate the following expression: $$\Phi(U,V)=\mathrm{trace}\...
-1
votes
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 ...
0
votes
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(...
1
vote
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
votes
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
votes
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
vote
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 ...
0
votes
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{\...
0
votes
0answers
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 ...