# Questions tagged [vector-analysis]

Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

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### Sketching and describing regions determined by inequalities in the complex plane

Can anyone teach me how to answer this question? Sketch the following sets in the complex plane $\mathbb{C}$ and determine whether they are open, closed, or neither; bounded; connected. Briefly state ...
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### how to explain a case when gradient vector does not point to the steepest direction

Please refer to attached diagram. Since the density of the projection of the contour lines is low at the middle part, z rises slower in the middle. However, since gradient vectors are orthogonal to ...
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The matrix problem is given by: $$y=M^T * A \circ B$$ where $M\in\mathbb{R}^{n}\times\mathbb{R}^m, A,B\in \mathbb{R}^{n}\times\mathbb{R}^n$ $\circ$ is Hadamart (element-wise) product and $*$ is matrix ...
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### What does the parentheses mean in vector calculus?

What does $\vec{a}(\vec{b} + \vec{c}) means?$ Is it $\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$ or is it the cross product $\vec{a}\times\vec{b}+\vec{a}\times\vec{c}$?
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### Initial margin covariance matrix

I am trying to replicate the following example in order to calculate IM : https://www.clarusft.com/isda-simm-in-excel-equity-derivatives/?amp=1 I am stuck at the last step (impossible to find the ...
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### Finding a specific function on $\mathbb{R}^3$ with some symmetry properties and partial knowledge of the function

I've been stumped with a problem for some time and thought I'd try my chance here. I am looking for a function $\vec{a}:\mathbb{R}_+^3\to \mathbb{R}_+^3$ such that if $\vec{a}=(a_1,a_2,a_3)$, then the ...
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### Using integration by parts to show $\int_{\Sigma} |\nabla^N_{\Sigma} X|^2 = - \int_{\Sigma} \langle X, \Delta^N_{\Sigma} X \rangle$

I'm trying to work through a derivation of the stability operator from minimal surface theory. Suppose $\Sigma^k$ is a minimal submanifold of $\mathbb{R}^n$, and suppose $X$ is a normal vector field ...
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### Why is the gradient a linear operator? [closed]

ie. why is the following true? If $f$ and $g$ are two real-valued functions differentiable at the point $a \in \mathbb{R}^n$, and $\alpha$ and $\beta$ are two constants, then $\alpha f + \beta g$ is ...
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### Finding a formula for an inverse square field in terms of a position of object in field

Say we have: $F(x,y) = \dfrac{ϵq_1q_2}{|r|^2} \hat{u}$ I am attempting to find the formula in terms of the position of the charge $q_2$ in $(x,y)$, while leaving everything else constant. (thanks ...
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### Can the "gradient is perpendicular to the level set" property only hold for one particular point?

A famous theorem says that gradient is perpendicular to the level set. Indeed, wiki says If the function $f$ is differentiable, the gradient of $f$ at a point is either zero, or perpendicular to the ...
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### Practice Problems for Revising Basic Results from Analysis

Does anybody have a good resource with worked solutions for quick revision of some basic results from analysis? I am self-studying a maths course on Fixed-Point Theorems (Course notes here) over the ...
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### Toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates

I am trying to understand some aspects of the toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates. We have a divergence-free ('solenoid') vector field $u$ on a compact ...
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### Applying Divergence Theorem or an analogue to a Closed Surface Integral

Consider $\oint_S({\nabla\cdot\vec{F}})dA$ (the closed surface integral of the divergence of a vector field). Is there a theorem which rewrites this integral as:$$\oint_B{(\vec{F}\cdot \hat{n})dl}$$ ...
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### Orthonormal basis given by the integral of a vector field along a curve orthogonal to the tangent

A friend and I are trying to understand the construction on section 3.2 of this paper by Bernatzki and Ye. Suppose $\gamma$ is a smooth simple closed curve in $\mathbb{R}^3$. Then this paper claims ...
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### Calculation of $ga:\triangledown^2 f$

I am reading a online note about the adjoint operator. One of the example uses multivariate calculus computation involving the gradient operator $\triangledown$. The equation on the note is \begin{...
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### How to calculate gradient of dot product?

How to calculate gradient of $\langle \theta, X^TX\theta \rangle - 2 \langle \theta, X^T y \rangle + \langle y,y \rangle$ over $\theta$? The answer is $2 X^T X \theta - 2 X^T y$. But I do not ...
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### Manipulation of delta (as in change) quantities

I am an undergrad physics student working through 4ed of Griffith's Electrodynamic book. This question is purely mathematical, however. Do not fret. In Griffith's (pg. 198), these equivalences are ...
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### Applying Reynold's Transport Theorem on expanding sphere to differentiate under the integral sign with varying limit

I'm working through the proof of the mean value inequality (1.15) of Colding-Minicozzi's A Course on Minimal Surfaces, and I'm stuck on this subproblem. Let $\Sigma$ be a $k$-dimensional minimal ...
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I am stuck in understanding how to proceed about the critical points of this function: $$f(x, y) = x^2y - xy - x$$ So I did the following: \nabla f = (0, 0) \longrightarrow \begin{cases} 2xy - y = 0 ...
Vector calculus is relatively new to me so I have a little trouble understanding double integrals intuitively. The question is esssentially this: Calculate the surface integral of $\vec v=x^2 \hat j$...