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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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Is the product of exponentiated elliptic curve basis elements invariant under FFT of scalars?

I am working with an elliptic curve defined over a finite field $\mathbb{F}_p$ and have a basis set of points ${g_0, g_1, \ldots, g_n}$. When I perform the FFT on these points, I obtain a new basis ${...
Nerses Asaturyan's user avatar
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Derivative of a matrix w.r.t a scalar [closed]

I have a notation in the Frobenius form, which is denoted as $\||\mathbf{Y}-\alpha\mathbf{A}\mathbf{B}\||_F^2$. I want to find the derivative of the scalar $\alpha$ and update this parameter with ...
gy L's user avatar
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Gradient of a sum of logarithms

Let the scalar field $f : {\Bbb R}_{>0}^n \to {\Bbb R}$ be defined by $$ f ({\bf x}) = \sum_{i=1}^{n}\ln x_i $$ and find the gradient $\nabla f$. I am new to matrix and vector derivatives. I am ...
Harry556's user avatar
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Is the scalar field ${\bf A} \mapsto \mbox{Tr} \left( {\bf A} {\bf B} {\bf A}^\top {\bf C} \right)$ convex?

Let ${\bf A}, {\bf B}, {\bf C}$ be $d \times d$ (symmetric) positive semidefinite matrices. Let $\mbox{Tr}$ denote the trace operator. Is the scalar field ${\bf A} \mapsto \mbox{Tr} \left( {\bf A} {\...
sabo's user avatar
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Minimizing a line integral in 2 dimensions : $\inf \int_{(a,b)}^{(c,d)} f(r(t)) |r'(t)| dt$

Let $x,y,z$ be real. Consider a scalar field $$z = f(x,y)$$ More specific; $f(x,y)$ is a (given) real polynomial in $x,y$ of degree at most $5$ such that For all $x,y$ $$f(x,y)> 0$$ For a given ...
mick's user avatar
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5 votes
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Scalar integrals in higher dimensions

The thing I want to do The typical vector calculus course defines: A bunch of integrals of vector fields in $\mathbb R^2$ and $\mathbb R^3$: line integrals of a vector field along a curve, flux ...
Misha Lavrov's user avatar
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Find the Scalar Tangential component of acceleration

Question : r(t) = 3sint i + 2 cost j - sin2t k at t=π/2 Find Scalar Tangential component of acceleration. Answer: Given, r(t) = 3sint i + 2 cost j - sin2t k Velocity,V = r'(t) = 3cost i - 2sint j - ...
Marvel's user avatar
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Regarding the change of a scalar field in a cone

Suppose we have a scalar field $T$ and we take a point in space, say $(\alpha,\beta,\gamma)$. The direction of the gradient $\nabla T$ at $(\alpha,\beta,\gamma)$ gives us the direction along which the ...
Rajdeep Sindhu's user avatar
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Integration over a Cartesian product

Let $x_1,x_2\in\mathbb{R}^d$ and $p(x_1,x_2):\mathbb{R}^{d}\times\mathbb{R}^d\mapsto\mathbb{R}$ be a generic scalar function. Let $S\subseteq\mathbb{R}^d$ be a generic domain of integration. Assume ...
matteogost's user avatar
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The function is $F(x,y,z) = x^2y^2z$. Curve C is the intersection of the surfaces $z=2-x^2-y^2$ and $z=\sqrt{(x^2+y^2)}$. Calculate $\oint F ds$.

The function is $F(x,y,z) = x^2y^2z$. Curve C is the intersection of the surfaces $z=2-x^2-y^2$ and $z=\sqrt{(x^2+y^2)}$. Calculate $\oint_C F ds$. Attempt: Firstly, to define the term inside the ...
Vile's user avatar
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2 answers
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Tensor fields and scalar function pullbacks

For convenience, $(p,q)$ tensor fields on a differentiable manifold $M$ is defined to be the entire scalar field. On the other hand, in my textbook, the pullback of the $(p,q)$ tensor $T(x)$ at $x \...
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Understanding why $\underline{\nabla} \phi = (\underline{\nabla} f) \frac{d \phi}{df}$

I'm struggling to understand a step in my lecture notes. Given a scalar field $f: \mathbb{R}^n \to \mathbb{R}$ and a function $\phi: \mathbb{R} \to \mathbb{R}$, \begin{align} \underline{\nabla} \phi &...
Ryderr's user avatar
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Prove that $\vec{\nabla} r^n = n r^{n-2} \vec{\tilde{r}}$

I need to prove that $\vec{\nabla} r^n = n r^{n-2} \vec{\tilde{r}}$ I have a demonstration using spherical coordinates in my notebook, but I would like to know how to do it the standard way (cartesian ...
Alex Vergara's user avatar
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On the $\log \det$ of identity matrix plus a symmetric positive definite matrix

I am trying to learn some matrix differentiation, and came across example of calculating the derivative of $$f(X)=\log\det(X)$$ where the $X$ is a symmetric positive definite matrix. I came to the ...
Michael's user avatar
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3 answers
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Differentiation matrix notation

I am trying to differentiate $$ \mathbf{b} \mapsto \mathbf{(a+b)}^\intercal \mathbf{A(a+b)} $$ where $\mathbf{a}$ and $\mathbf{b}$ are $n \times 1$ vectors and $\mathbf{A}$ is an $n \times n$ ...
The User's user avatar
3 votes
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112 views

Can we classify all functions whose gradient is an eigenvector of the Hessian?

Let's treat the case of two dimensions, then we don't have freedom with the other eigenvector. Is there any classification of all smooth functions $f$ on $\mathbb{R^2}$ such that $\nabla f$ is an ...
Math Addiction's user avatar
1 vote
2 answers
140 views

Find the derivative of a diagonal matrix and norm

Find the derivative with respect to $X \in \mathbf{R}^{n \times p} $ of $$ \Phi(X) = \operatorname{Tr} \left( X^{\top} H(X) X \right) $$ where $H(X) := D(X) A D(X)$, where $A$ is symmetric and $$D(X) =...
Alaeddine Zahir's user avatar
3 votes
1 answer
104 views

Gradients of $(u, v) \mapsto \frac12 \left\| A - u v^T \right\|_{\text{F}}^2$ via the chain rule

Given the matrix $A \in {\Bbb R}^{n \times m}$, let the scalar field $f : {\Bbb R}^n \times {\Bbb R}^m \to {\Bbb R}_0^+$ be defined by $$ f (u, v) : = \frac12 \left\| A - u v^T \right\|_{\text{F}}^2 $$...
ek_q_t's user avatar
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What is the Hessian of $x \mapsto\log \det \left( A^T A + R^T \operatorname{diag}(x)^{-1} R \right)$?

This is a follow-up to a previous question I asked regarding the hessian of a similar log determinant. The log determinant I am considering is given by $$ L(\vec{x}) = \log \det \left( A^T A + R^T D_x^...
Jonathan Lindbloom's user avatar
2 votes
3 answers
245 views

Second order derivative of $f(x):=\frac{1}{2} ⟨x,Ax⟩$

Let $A=\left(A_{i j}\right)$ be an $n \times n$ symmetric matrix, and define the function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ as $$ f(x):= \frac{1}{2} ​ ⟨x,Ax⟩ $$ Using the definition, ...
Allison's user avatar
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146 views

Gradient of ${\bf z} \mapsto- \frac12 \operatorname{tr} \left( {\bf A} \left( {\bf Y} - {\bf x} {\bf z}^\top \right) {\bf B} ( \cdot )^\top \right)$

Given $3 \times 3$ symmetric matrix $\bf A$ and $5 \times 5 $ symmetric matrix $\bf B$, let the scalar field $f : \Bbb R^5 \to \Bbb R$ be defined by $$ f ({\bf z}) := - \frac12 \operatorname{tr} \...
parx's user avatar
  • 11
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0 answers
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Find the constants $a,b,c$ such that the directional derivative is maximum on the direction of $\vec{u}$

Let $f:\mathbb{R}^3\rightarrow \mathbb{R}$ be the function defined as $f(x,y,z)=ax^2y+by^2 z+cz^2 x$. Find the constants $a,b,c$ such that in the point $(1,1,1)$, the directional derivative is maximum ...
Aley20's user avatar
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Derivative of $\frac{\partial f(X^{\top}X)}{\partial X}$

The task is to prove that for any matrix $X$ and differentiable scalar function $f$, the following holds: $$ \frac{\partial f(X^\top X)}{\partial X} = 2{X}\frac{\partial f(X^\top X)}{\partial (X^\top ...
That1WasTaken's user avatar
1 vote
1 answer
88 views

Gradient of $(x,y) \mapsto y^\top A(x) y$

Given a differentiable $A: \mathbb{R}^n \to \mathbb{R}^{m \times m}$, let $f: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ be a scalar field defined by $$f(x,y) := y^\top A(x) y$$ Can the ...
Trb2's user avatar
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177 views

Mean-value theorem for function of matrices

Consider a twice differentiable function $f: {\Bbb R}^{d \times d} \to {\Bbb R}$ with bounded derivatives $$ \left | \frac{\partial^2 f}{\partial X_{kl} \partial X_{ij}} \right | \leq D $$ I am ...
atownmath's user avatar
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54 views

Is the gradient of a function $f : \mathbb{R}^2 \to \mathbb{R}^1$ the best $1$-dimensional approximated linear transformation at a point $v$?

I'm trying to get a better intuition regarding Jacobians. I think I have a decent understanding of gradients and what their directions mean, but I'm trying to connect all the dots for the ...
Caj's user avatar
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2 votes
1 answer
208 views

Gradient of $C \mapsto\frac{1}{2}\left\lVert CA - BC \right\rVert_F^2$

Given the matrices $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{m \times m}$, let the scalar field $f : \mathbb{R}^{m \times n} \to \mathbb{R}$ be defined by $$ f(C) := \frac{1}{2}\left\...
user8469759's user avatar
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1 vote
2 answers
77 views

Gradient of a function w.r.t. matrices

Let $ \varphi : \mathbb{R}^m \times \mathbb{R}^{m \times n} \times \mathbb{R}^{n \times m} \to \mathbb{R} $ be defined by $$ \varphi(x, A, B) = \left \| \left( x^T A B \right)^T - x \right \|^2 $$ ...
Rabist's user avatar
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2 answers
122 views

How do you find the scalar field of the vector field $F=-y\hat{i}+x\hat{j}$

I know that: $F=\nabla f = \frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial y}\hat{j}+\frac{\partial f}{\partial z}\hat{k}$ So in cases like: $F=2x\hat{i}+2y\hat{j}$ I just integrate ...
mEXsACHINE's user avatar
3 votes
3 answers
132 views

Derivative of $\varphi({\bf X}) = \sum_{i=1}^n \lambda_i({\bf X}) \log \lambda_i({\bf X})$

Let $\mathbb{S}_+^n$ denote the set of $n \times n$ symmetric positive definite matrices. Let scalar field $\varphi : \mathbb{S}_+^n \to \Bbb R$ be defined by $$\varphi({\bf X}) := \sum_{i=1}^n \...
The Limit Does Not Exist's user avatar
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1 answer
93 views

Stuck while differentiating $x\mapsto\|x\|^2 + \lambda{\|Ax - b\|}^2$

I want to differentiate the following equation $$ x \mapsto \|x\|^2 + \lambda{\|Ax - b\|}^2 $$ where the real symmetric matrix $A$ is not invertible. I have decomposed $A$ using the spectral theorem, ...
julie's user avatar
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3 votes
2 answers
117 views

Show $\nabla f=A\nabla g$ by chain rule

Let $A$ be a $2\times 2$ an invertible matrix, $f:\mathbb R^2\to\mathbb R$ be smooth, and define $g:\mathbb R^2\to \mathbb R$ by $$g(Ax)=f(x),$$ i.e., if I set the variable of $g$ as $y$, $g(y)=f(A^{-...
daㅤ's user avatar
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4 answers
193 views

Gradient of $X \mapsto \mbox{tr} \left(BXCX^TB^TBXCX^TB^T\right)$

Let us assume that \begin{equation} f(X)=\mbox{tr}\left(XCX^TXCX^T\right), \end{equation} in which $C\in\mathbb{R}^{r\times r}$ is a symmetric matrix, and $X \in \mathbb{R}^{r'\times r}$. From the ...
Math-Data's user avatar
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3 answers
110 views

Why is function $X \mapsto z^T X z$ linear?

From Boyd & Vandenberghe's Convex Optimization: $\textsf{Example 2.7}\;\;$ The positive semidefinite cone $\mathbf{S}_+^n$ can be expressed as $$ \bigcap_{z\not=0} \{ X \in \mathbf{S}^n \mid z^...
user3180's user avatar
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2 answers
98 views

Gradient of $x \mapsto (w^tx+b)^2$

I have a non-zero column vector w $\in \mathbb{R^2}$ and a scalar b $\in \mathbb{R}$, so it's a function $f: \mathbb{R^n} \to \mathbb{R}$ with this definition: $f(x) = (w^tx + b)^2$, where $x$ is a ...
Alexis's user avatar
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1 answer
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Gradient of $X \mapsto \frac1{\sqrt{\det (X)}}$

Let $\Bbb S_n^{++}$ denote the set of $n \times n$ symmetric positive definite matrices over $\Bbb R$. Let scalar field $f : \Bbb S_n^{++} \to \Bbb R$ be defined by $$ f (X) := \frac1{\sqrt{\det(X)}} $...
heyoka955's user avatar
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Gradient of quadratic scalar field $X \mapsto \mbox{tr} (XAX)$

Given $n\times n$ matrix $A$, find (with proof) the gradient $\nabla_X \mbox{tr} (XAX)$. The matrix $A$ does not depend on $X$. I know that the final answer is of the form: $$\frac{\partial \mbox{tr} ...
Mike's user avatar
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0 votes
2 answers
126 views

Direction of gradient unit vector at r

Consider the scalar field in two dimensions f (x,y) = x2 + 4y2. This field has a global minimum at r = (0, 0). Write a computer program in which, starting from the initial point r(0) = (4, 1.5), you ...
Lucas's user avatar
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2 votes
1 answer
118 views

Gradient of ${\bf x}^\top {\bf A}^{1/2} {\bf x}$ with respect to $\bf A$

How to calculate the gradient $\nabla_{\bf A} \left( {\bf x}^\top {\bf A}^{1/2} {\bf x} \right)$, where $\bf x$ is $N \times 1$ column vector and $\bf A$ is $N \times N$ symmetric positive matrix? The ...
Qiuyun.Zou's user avatar
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1 answer
92 views

Is this gradient inequality correct?

Let scalar field $f : \mathbb{R}^n \to \mathbb{R}$ be smooth, i.e., in $C^\infty (\mathbb{R}^n)$, and there exists $\bar{x}$ be a point on the line segment connecting $x_1$ and $x_2$. Is the following ...
Nichts__97's user avatar
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1 answer
60 views

If a scalar product is not specified on a vector space over an arbitrary field, is there a "standard" scalar product assumed?

In Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, by Jürgen Richter-Gebert we are asked to consider the three-dimensional vector space over an arbitrary field $\...
Steven Thomas Hatton's user avatar
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2 answers
585 views

Derivative of the determinant with respect to the vector

There is a vector $\boldsymbol{v}=[x, y, z]^T$ and a matrix $\boldsymbol{A}(x,y,z)$. How is the derivative of a determinant of a matrix with respect to a vector calculated, i.e.: $\frac{d|\boldsymbol{...
ayr's user avatar
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1 vote
2 answers
94 views

Criterion for globality of extreme point of a scalar field?

So, lets say we have $\DeclareMathOperator{\int}{int}$$\Omega \subset \mathbb{R}^n$, $ f:\Omega\rightarrow \mathbb{R}$ being continuous and partially differentiable on $\int\Omega$. Now there is $x_*\...
TiimmyTurner's user avatar
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1 answer
67 views

Gradient field of scalar field functions

The scalar field functions $s$ are defined in space by: $$s(x,y) = x^2y^2 + xy - z + C$$ How do I show that all functions have an identical gradient field and how do I calculate it?
Max_User1's user avatar
6 votes
0 answers
85 views

If the gradient of two functions are related by a matrix can we say anything about these two functions?

Say we have two functions $f, g: \mathbb{R}^n \rightarrow \mathbb{R}$ and we have a matrix $M \in \mathbb{R}^{n \times n}$ which is constant, i.e. not a function of $x$. Say we know that $\nabla f = M ...
Amar Shah's user avatar
1 vote
3 answers
612 views

First-order Taylor approximation of matrix function

Let $f : \Bbb C^{M \times N} \to \Bbb R_0^+$ be defined by $$ f(X) := \left\| X X^H - R \right\|_F^2 $$ I would like to find the first-order Taylor approximation of $f$. I am familiar with the vector ...
Duns's user avatar
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-2 votes
2 answers
139 views

What is the gradient of $x \mapsto \frac{1}{2}(x^2)^\top A(x^2)$?

Let matrix $A$ be sparse and symmetric but not semidefinite. Since I would like to use projected gradient descent, I must find the gradient of $x \mapsto \frac{1}{2}(x^2)^\top A(x^2)$, where $x^2 = \...
someone random's user avatar
0 votes
1 answer
56 views

When is $\nabla\cdot({\bf a}\cdot{\bf r})({\bf b}\cdot{\bf r})\,{\bf F}({\bf r})=0$ for constant vectors ${\bf a}$ and ${\bf b}$?

When is the assertion that $$\nabla\cdot({\bf a}\cdot{\bf r})({\bf b}\cdot{\bf r})\,{\bf F}({\bf r})=0$$ for constant vectors ${\bf a}$ and ${\bf b}$ and a everywhere-divergenceless ${\bf F}$ true? ...
Ice Tea's user avatar
  • 445
2 votes
1 answer
65 views

Show that there is a point $x_0\in\Bbb{R}^n$ such that $Df(x_0)=0$

Let $f : \Bbb{R}^n \to \Bbb{R}$ be a $C^1$ function such that $$\lim\limits_{\|x\|\to\infty}f(x)=0$$ Show that there is a point $x_0 \in \Bbb{R}^n$ such that $Df(x_0)=0$. I'm struggling to finish ...
Daniil's user avatar
  • 1,677
0 votes
0 answers
51 views

Differentiability of $g(x,y) := f \left(\sqrt{x^{2} + y^{2}}\right)$

Suppose the function $f : \Bbb R \to \Bbb R$ is continuously differentiable and define another function $g$ as $$g(x,y) := f \left(\sqrt{x^{2} + y^{2}}\right)$$ Under what condition is $g$ ...
krianaaa's user avatar
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