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.
385 questions
3 votes
1 answer
164 views
Extrema of $H \mapsto \mbox{tr} \left( H^2 X \right) - \left(\mbox{tr} (HX) \right)^2$ where $H$ is traceless and $\mbox{tr} \left(H^2\right)=1$
Assume that $X$ is an $n\times n$ positive definite matrix with $\operatorname{tr}(X) = 1$. Consider the function $$ f(H) := \operatorname{tr} \left(H^2 X\right) - \operatorname{tr} \left(HX\right)^2 $...
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2 votes
1 answer
88 views
A PDE that characterizes linear functions?
All linear scalar fields $f : \mathbb R^n \to \mathbb R$ are solutions to the following PDE: $$\langle \nabla f_p,\ p \rangle = f(p)$$ for every $p \in \mathbb R^n.$ (A more general statement can be ...
- 1,512
2 votes
3 answers
263 views
Gradient of $X \mapsto \operatorname{tr} \left( \left(X X^T \right)^n \right)$
Assume $X$ is a real matrix. Is there a way to compute $\frac{d}{dX}tr[(XX^T)^n]$? For $n=1$, it is a known result that $\frac{d}{dX} tr[XX^T]=2X$. What about higher $n$? EDIT: There seem to be ...
- 349
0 votes
1 answer
113 views
Gradient of the linear scalar field $B \mapsto w'B's$ [duplicate]
I want to get the derivative of function $f(B) = w'B's$ with respect to $B$, where $w,s$ are column vectors, then $df = w'(B+dB)'s - w'B's = w'dB's = s'(dB)w$. I know the answer is $sw'$, but I could ...
- 79
2 votes
2 answers
119 views
Gradient of $W \mapsto (\mathbf{t} - W^\top \mathbf{x})^\top A (\mathbf{t} - W^\top\mathbf{x})$
Let $$f(W) = (\mathbf{t} - W^T\mathbf{x})^TA(\mathbf{t} - W^T\mathbf{x})$$ I expanded to get: $$f(W) = \mathbf{t}^TA\mathbf{t} - \mathbf{t}^TAW^T\mathbf{x} - \mathbf{x}^TWA\mathbf{t} + \mathbf{x}^TWAW^...
- 661
3 votes
1 answer
190 views
Tricky chain rule for matrices
Note: This is my first time asking a question on Mathematics SE and I apologise if my question is not well-formulated. I am trying to compute the derivative of a log likelihood function with respect ...
2 votes
0 answers
63 views
How to Determine the Weight Function for a Given Set of Orthogonal Polynomials?
I am studying orthogonal polynomials and their associated weight functions, and I am trying to understand how one can determine the underlying weight function $w(x)$ for a given set of orthogonal ...
2 votes
0 answers
137 views
Question regarding proof of second-order Taylor formula for scalar fields
The following theorem can be found in section 9.10 of Apostol's Calculus Vol. II (2nd edition). Theorem: Let $f:S\subset\mathbb{R}^n\to \mathbb{R}$ and $\varepsilon>0.$ If the second order partial ...
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0 votes
0 answers
59 views
Understanding section 8.16 of Calculus Vol. II by Apostol
I'm reviewing section 8.16 of Calculus Vol.II, regarding Level sets and tangent plane. Let us consider a scalar field $f:S\subset\mathbb{R}^n\to \mathbb{R}$, the level set is defined as all those ...
- 1,318
-1 votes
1 answer
166 views
Are "scalar field" and "scalar-valued function" synonymous?
Is a "scalar field" synonymous with a "scalar-valued function"? If not, is one a sub-type of the other; in what way? It seems that scalar fields must be continuous and ...
- 1,239
0 votes
1 answer
77 views
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 ...
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0 votes
0 answers
185 views
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 ...
- 444
1 vote
1 answer
114 views
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} {\...
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1 vote
0 answers
104 views
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 ...
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5 votes
1 answer
375 views
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 ...
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