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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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 ...
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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 &...
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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 ...
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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 ...
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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 $n \times n$ ...
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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 ...
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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 $$D(X) =...
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Gradients of $(u, v) \mapsto \frac12 \left\| A - u v^T \right\|_{\text{F}}^2$ via the chain rule
Given the matrix $A \in {\Bbb R}^{n \times m}$, let the scalar field $f : {\Bbb R}^n \times {\Bbb R}^m \to {\Bbb R}_0^+$ be defined by
$$ f (u, v) : = \frac12 \left\| A - u v^T \right\|_{\text{F}}^2 $$...
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What is the Hessian of $x \mapsto\log \det \left( A^T A + R^T \operatorname{diag}(x)^{-1} R \right)$?
This is a follow-up to a previous question I asked regarding the hessian of a similar log determinant. The log determinant I am considering is given by
$$
L(\vec{x}) = \log \det \left( A^T A + R^T D_x^...
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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, ...
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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
$$ f ({\bf z}) := - \frac12 \operatorname{tr} \...
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Find the constants $a,b,c$ such that the directional derivative is maximum on the direction of $\vec{u}$
Let $f:\mathbb{R}^3\rightarrow \mathbb{R}$ be the function defined as $f(x,y,z)=ax^2y+by^2 z+cz^2 x$. Find the constants $a,b,c$ such that in the point $(1,1,1)$, the directional derivative is maximum ...
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Derivative of $\frac{\partial f(X^{\top}X)}{\partial X}$
The task is to prove that for any matrix $X$ and differentiable scalar function $f$, the following holds:
$$
\frac{\partial f(X^\top X)}{\partial X} = 2{X}\frac{\partial f(X^\top X)}{\partial (X^\top ...
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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 ...
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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 ...
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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 ...
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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
$$ f(C) := \frac{1}{2}\left\...
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Gradient of a function w.r.t. matrices
Let $ \varphi : \mathbb{R}^m \times \mathbb{R}^{m \times n} \times \mathbb{R}^{n \times m} \to \mathbb{R} $ be defined by
$$ \varphi(x, A, B) = \left \| \left( x^T A B \right)^T - x \right \|^2 $$
...
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How do you find the scalar field of the vector field $F=-y\hat{i}+x\hat{j}$
I know that:
$F=\nabla f = \frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial y}\hat{j}+\frac{\partial f}{\partial z}\hat{k}$
So in cases like:
$F=2x\hat{i}+2y\hat{j}$
I just integrate ...
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Derivative of $\varphi({\bf X}) = \sum_{i=1}^n \lambda_i({\bf X}) \log \lambda_i({\bf X})$
Let $\mathbb{S}_+^n$ denote the set of $n \times n$ symmetric positive definite matrices. Let scalar field $\varphi : \mathbb{S}_+^n \to \Bbb R$ be defined by
$$\varphi({\bf X}) := \sum_{i=1}^n \...
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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, ...
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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^{-...
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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 ...
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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
$$ \bigcap_{z\not=0} \{ X \in \mathbf{S}^n \mid z^...
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Gradient of $x \mapsto (w^tx+b)^2$
I have a non-zero column vector w $\in \mathbb{R^2}$ and a scalar b $\in \mathbb{R}$, so it's a function $f: \mathbb{R^n} \to \mathbb{R}$ with this definition: $f(x) = (w^tx + b)^2$, where $x$ is a ...
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Gradient of $X \mapsto \frac1{\sqrt{\det (X)}}$
Let $\Bbb S_n^{++}$ denote the set of $n \times n$ symmetric positive definite matrices over $\Bbb R$. Let scalar field $f : \Bbb S_n^{++} \to \Bbb R$ be defined by $$ f (X) := \frac1{\sqrt{\det(X)}} $...
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Displacement field, vector field or scalar field? Nabla operator applied to displacement field.
I have two questions both related.
Firstly, is the displacement field a scalar field or a vector field? It is my understanding that because each point in the field has a direction given by ux and uy, ...
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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:
$$\frac{\partial \mbox{tr} ...
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Direction of gradient unit vector at r
Consider the scalar field in two dimensions f (x,y) = x2 + 4y2. This field has a global minimum at
r = (0, 0). Write a computer program in which, starting from the initial point r(0) = (4, 1.5), you ...
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Gradient of ${\bf x}^\top {\bf A}^{1/2} {\bf x}$ with respect to $\bf A$
How to calculate the gradient $\nabla_{\bf A} \left( {\bf x}^\top {\bf A}^{1/2} {\bf x} \right)$, where $\bf x$ is $N \times 1$ column vector and $\bf A$ is $N \times N$ symmetric positive matrix?
The ...
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Is this gradient inequality correct?
Let scalar field $f : \mathbb{R}^n \to \mathbb{R}$ be smooth, i.e., in $C^\infty (\mathbb{R}^n)$, and there exists $\bar{x}$ be a point on the line segment connecting $x_1$ and $x_2$. Is the following ...
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If a scalar product is not specified on a vector space over an arbitrary field, is there a "standard" scalar product assumed?
In Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, by Jürgen Richter-Gebert we are asked to consider the three-dimensional vector space over an arbitrary field $\...
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Derivative of the determinant with respect to the vector
There is a vector $\boldsymbol{v}=[x, y, z]^T$ and a matrix $\boldsymbol{A}(x,y,z)$.
How is the derivative of a determinant of a matrix with respect to a vector calculated, i.e.:
$\frac{d|\boldsymbol{...
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Criterion for globality of extreme point of a scalar field?
So, lets say we have $\DeclareMathOperator{\int}{int}$$\Omega \subset \mathbb{R}^n$, $ f:\Omega\rightarrow \mathbb{R}$ being continuous and partially differentiable on $\int\Omega$. Now there is $x_*\...
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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?
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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 ...
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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 ...
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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 = \...
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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? ...
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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 ...
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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 $g$ ...
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Parametrising Intuition of Plane
This might be a bit too "hand-wavy" for this forum but here it goes:
Generally the problem is this, I wish to create a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ with some properties ...
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Remap 3D function onto a plane?
I'm working on a design project in nTopology which is an implicit representation design software. So geometry objects are represented as implicit fields defined by continuous functions.
Bodies are ...
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Gradient of $\mbox{tr} \left (X^T X \right)$
My goal is to compute
$$\frac{\mathrm{d} \operatorname{tr}\left(\mathbf{X}^{T} \mathbf{X}\right)}{\mathrm{d} \mathbf{X}}$$
Following the common way of approaching vector/matrix differentiation, I ...
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How to find the directional derivative at every point?
Given a function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $$f(x,y) = \begin{cases} \left( x^2 + y^2 \right) \cos \left(\frac{1}{x^2 + y^2} \right) & \text { if } (x,y) \neq (0,0)\\ 0 & \...
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Surface integral of hemisphere
In a scalar field I need to calculate the surface integral of this:
$$\iint_{\Sigma}\frac{d \sigma}{\sqrt{x^2+y^2+(z+R)^2}}$$ with $\Sigma$ the upper half of the sphere $x^2+y^2+z^2=R^2$
The formula ...
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Partial derivatives in scalar field taylor expansion
$\newcommand{\v}[1]{\mathbf{#1}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\dd}[1]{\mathrm{d}#1}$
In our lecture notes we derived the following formula for the Taylor expansion of a scalar ...
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What makes complex numbers so dominant among two-dimensional scalar fields?
Complex numbers are a two-dimensional field. But other 2d fields can be defined. Edit: That is incorrect, no other 2d scalar field can be defined.
For example, the standard vector addition and the ...
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Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$
Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$ with $\mathbf{A}\in\mathbb{R}^{M\times N}$ and $\mathbf{X}\in\mathbb{R}^{N\times D}$, and ...
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