# 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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### Functions defined by the second derivative

I am looking for classes of differentiable functions that are characterized by their second derivative. For example: convex functions: function $f$ is convex iff its Hessian, $\nabla^2 f$, is ...
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### Calculate the gradient of a linear scalar field [duplicate]

I am trying to calculate the following gradient $$\nabla_{\mathbf{X}} \left( \mathbf{a}^{T} \mathbf{X} \mathbf{a} \right)$$ where I am using the convention that $\mathbf{a}$ is a column vector. I am ...
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### Finding irrotational and scalar potential [closed]

Show that the vector $F$ is irrotational and hence find it's scalar potentialenter image description here
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Let $f: \mathbb{R}^n \to \mathbb{R}$. Let $M$ be positive definite. Is it always true that $f(x)$ increases in the $M \nabla f$ direction, where $\nabla f$ is the direction of steepest ascent (the ...
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### How to differentiate $g(X)=\operatorname{tr}\left(X^{-1}\right)$? [duplicate]

Let $X$ be a square invertible $n \times n$ matrix. Calculate the derivative of the following function with respect to X. $$g(X)=\operatorname{tr}\left(X^{-1}\right)$$ I'm stumped with this. As when ...
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### Matrix derivative of $\mathrm{tr}((I+X^{-1})^{-1})$

I'm trying to calculate the derivative of $\mathrm{tr}((I+X^{-1})^{-1})$ with respect to $X$. By some sort of a chain rule, I believe this should be $X^{-1}(I+X^{-1})^{-2}X^{-1}$. However, I'm having ...
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### Why in $GF(7)$ there are no nonzero elements satisfying $a^2 + b^2 = 0$?

Why in $GF(7)$ there are no nonzero elements satisfying $a^2 + b^2 = 0$? how can I know this by a quick method without calculations? Is there a theorem or something that can help here? $GF(7)$ is the ...
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### Distribution of multiplication over addition in a field.

I am studying Linear algebra from a book called "The Linear Algebra a Beginning Graduate Student Ought to Know " by Jonathan S. Golan. And in the definition of a field the book mentioned that the ...
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### If scalar field $f(x,y,z)$ is differentiable then the set of point satisfying $f(x,y,z)=c$ ($c$ is constant) is smooth surface?

Let $f(x,y,z), x,y,z\in \mathbb{R}$ is a scalar field. Could you prove that if $f$ differentiable then the set of points $(x,y,z)$ sastifying $f(x,y,z)=c$ ($c$ is constant) is a smooth surface? The ...
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### The conjugate of a scalar is the same scalar in a matrix-scalar multiplication?

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section 2.4, Page 97, I thought the equation is right as followed: $(cA)^{*} = cA^{*}$ But answer ...
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### Under what conditions are partial derivatives continuous?

For any continuous scalar and vector fields which doesn't contain any singular points, under what conditions are their partial derivatives continuous?
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### Matrix derivative of $Tr(A\log(X))$

I'm trying to work out the derivative of $Tr(A\log(X))$ with respect to $X$. Assume $X$ is positive so the $\log$ is well defined. I know that $$Tr(A\log(X)) = A^\dagger: \log(X)$$ but what I ...
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### Why is this notation equal to its transpose?

In my econometrics textbook, I have this step which is not clear to me: \begin{align} S &= e'e \\ &= (y-W\beta)'(y-W\beta) \\ &= \underbrace{y'}_{1\times T} \ \underbrace{y}_{T\times 1} ...