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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Can someone solve this problem on linear algebra? [closed]

This problem on linear algebra is from the past entrance exam for CS of the University of Tokyo grad school. I think I got the first question; x+y+z != 1 where n4 = xn1 + yb2 + zn3 but from the next ...
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The reason for curl free

I wonder about the reason for the idea of this, would you mind explain for me this can happen in mathematics. Thank you !
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Why is the magnitude of the cross product equal to the parallelogram spanned by the two vectors?

Given vectors $a, b \in \mathbb{R}^3$, the cross (vector) product of $a \times b = c$ is defined as a vector orthogonal to both $a$ and $b$ such that the right hand rule is satisfied, with the ...
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Differentiability of a function and existence of directional derivatives

I have a problem. In the book of Classical Elemental Analysis of Hoffman - Marsden assure that the function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x,y) = \frac{xy}{x^2+y}$ if $x^2 \neq -y$ and ...
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region above the cone and inside the sphere

I've come across a problem where \begin{array}{c} \mbox{I have to compute the volume of the region} \\ \mbox{above the cone}\ z^2 = \sqrt{x^2 + y^2}\ \mbox{and inside the sphere}\ x^2 + y^2 + z^2 = 1 \...
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Taylor Expansion of a vector-valued function with 2 vectors as input

Let a function $f(x,u): \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$. I wonder how to expand it around $(x_n, u_n)$. For the time being, keeping it only up to the first order is enough ...
1 vote
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Prove that in the canonical basis $\mathrm{grad}\,f = \sum_{j=1}^{n} \frac{\partial f}{\partial x_j}\,e_i$.

Reading Do Carmo I found this exercise: Given a differentiable function $f:\mathbb{R}^n \to \mathbb{R}$ the vector field (gradient) is defined by $$\langle \mathrm{grad}\,f(p) \rangle =df_p(u)$$ ...
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Matrix algebra: Kronecker product and integrals involving exponentials

I have been trying to figure out how the following might work. For matrices (of conformable dimensions) $A$ and $B$, and an identity matrix $I$, we have \begin{equation*} -(A\otimes I+ I\otimes B)^{-1}...
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Vector analysis text book.

I saw in the post orthogonal to the level curve a few screenshots from a textbook, and I would like to know if by any chance someone knows that text book's name, I tried searching the images in Google ...
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Understanding the Equation for Unit Vectors in Curvilinear Coordinate Systems

I am trying to understand the derivation of divergence in curvilinear coordinate systems. I stumbled upon this equation and it seems enigmatic to me. I know that $\mathbf{e}_{u_i}$ are the unit ...
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Abuse of Leibniz notation on differential n-forms

Leibniz notation is notorious for being abused. e.g. Chain rule: $$\frac{dy}{dt}= \frac{dy}{dx}\frac{dx}{dt}$$ Inverse Function Theorem: $$\frac{dy}{dx} = \left(\frac{dx}{dy}\right)^{-1}$$ Separation ...
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1 vote
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Understanding the Relationship Between Cross Product Components and Differential Forms on a Membrane

I am struggling with a differential geometry and vector calculus concept involving the relationship between the cross-product components and differential forms. The specific context is a response ...
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Help with proof of Curl Double Product identity using Geometric Algebra. Most things seem to fall in place, but having a few issues.

So I'm pretty new to GA/Clifford Algebras, but it's been fairly interesting so far. I figured I'd try to prove some basic vector calculus identities with it, just to help me get my bearings. I decided ...
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Can you use the interior product to calculate flux?

If I have a vector Field F and a 2D surface $\Sigma$ in $\mathbb{R}^3$ would $F \lrcorner dx \land dy \land dz= F_1 dy\land dz + F_2dz\land dx + F_3dx\land dy$ If this is the case, could one say that ...
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