# 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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### Why is gradient the direction of steepest ascent?

$$f(x_1,x_2,\dots, x_n):\mathbb{R}^n \to \mathbb{R}$$ The definition of the gradient is $$\frac{\partial f}{\partial x_1}\hat{e}_1 +\ \cdots +\frac{\partial f}{\partial x_n}\hat{e}_n$$ which is a ...
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Consider the vector field $$\vec{\mathrm{F}} = \frac{\hat{\mathrm{r}}}{r^{2}},$$ then the divergence of this field is: $$\vec{\nabla}\cdot\left(\frac{\hat{\mathrm{r}}}{r^{2}}\right) = 4\pi\delta^{3}(\... 26 votes 3 answers 21k views ### Anti-curl operator It is known that if a vector field \vec{B} is divergence-free, and defined on \mathbb R^3 then it can be shown as \vec{B} = \nabla\times\vec{A} for some vector field A. Is there a way to find ... • 859 34 votes 6 answers 178k views ### Surface Element in Spherical Coordinates In spherical polars,$$x=r\cos(\phi)\sin(\theta)y=r\sin(\phi)\sin(\theta)z=r\cos(\theta)$$I want to work out an integral over the surface of a sphere - ie r constant. I'm able to derive ... • 832 23 votes 2 answers 56k views ### Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory In order to prove the following identity:$$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$Instead of checking this by brute force, Landau writes thr product of ... • 2,585 23 votes 3 answers 6k views ### Do the BAC-CAB identity for triple vector product have some intepretation? As in the title, I was wondering if the formula:$$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$for \mathbb R ^3 cross product has some geometrical interpretation. I've recently seen a proof (from ... • 2,150 0 votes 2 answers 1k views ### Deriving the gradient operator in spherical coordinates This is a sort of problem where I know what to do but do not completely understand what I am doing. I have been taught how to derive the gradient operator in spherical coordinate using this theorem$$\...
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Could someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$? $$\nabla\|x\|_2^2 = 2x$$
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### Gradient of $a^T X b$ with respect to $X$

How can I find the gradient of the term $a^TXb$ where $X$ is a $n \times m$ matrix, and $a$ and $b$ are column vectors. Since the gradient is with respect to a matrix, it should be a matrix. But I ...
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### $C^1$ function on compact set is Lipschitz

Let $\emptyset\ne A\subset\mathbb{R}^n$ be open and let $f \in C^1 (A, \mathbb{R})$ be a function. Let $\emptyset\ne K\subset A$ be compact and convex. I want to prove that $f$ is Lipschitz on $K$; ...
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### Gradient of a dot product

The wikipedia formula for the gradient of a dot product is given as $$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$ However, I also ...
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### Proof for the curl of a curl of a vector field

For a vector field $\textbf{A}$, the curl of the curl is defined by $$\nabla\times\left(\nabla\times\textbf{A}\right)=\nabla\left(\nabla\cdot\textbf{A}\right)-\nabla^2\textbf{A}$$ where $\nabla$ is ...
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### Proof that gradient is orthogonal to level set

When we proved the gradient of a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is orthogonal to the level sets of the function $f(\vec{x}) = c$ for some constant $c$, my professor was quite explicit ...
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### Divergence of matrix-vector product

Suppose that $A$ is a matrix field and that $v$ is a vector field. What is the divergence of the matrix-vector product $A \cdot v$, which is a vector field?
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### Rotating vector functions

Suppose you have a vector function of space: $\vec{E}(x,y,z)=Ex(x,y,z)\hat{x}+Ey(x,y,z)\hat{y}+Ez(x,y,z)\hat{z}$. Suppose now you want to rotate the whole vector function by using a unitary rotation ...
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### Del operator in Cylindrical coordinates (problem in partial differentiation)

I am currently reviewing basic vector analysis and trying to understand every single detail, however, I got stuck in some derivation. What I want to show is the following: Given the del operator (...
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### What exactly are pseudovectors and pseudoscalars? And where could I read about them?

I can't find good information on the internet. In my mathematical physics class the definition of a vector was given as: That object with magnitude and direction which doesn't change under ...
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### Why is the determinant of the Jacobian the change of volume factor that comes from changing variables?

I can understand this through various examples found in the internet but I can't quite intuitively understand why the determinant of the derivative(in its most general form)-the Jacobian-gives the ...
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### Existence of Partial Derivatives Implies Differentiability [closed]

Doing some self-study. My textbook has this Theorem (see below). I understand it but was hoping for something shorter and more intuitive. Any thoughts?
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### prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation

I'm having some trouble using index notation to prove the identity $$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$ The closest I can get is by expanding ...
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### Why isn't the directional derivative generally scaled down to the unit vector?

I'm starting to learn how to intuitively interpret the directional derivative, and I can't understand why you wouldn't scale down your direction vector $\vec{v}$ to be a unit vector. Currently, my ...
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### Derivative of cross-product of two vectors

In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?
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### What is the intuition behind the unit normal vector being the derivative of the unit tangent vector?

I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction?
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### Existence and uniqueness of Stokes flow

What are the solution existence and uniqueness conditions for Stokes' flow? $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ Maybe you could ...
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### Product rule for scalar-vector product

Let $\mathbf F : \mathbb R^p \to \mathbb R^s$ and $\phi : \mathbb R^p \to \mathbb R$ be differentiable functions. Let the function $\mathbf G$ be defined as follows: \mathbf G : \mathbb R^p \to \...
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