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

Let us assume that $$f(X)=\mbox{tr}\left(XCX^TXCX^T\right),$$ 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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### 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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