# 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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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 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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### Closed surface integral of the surface's normal vector

Is it true that the surface integral over any closed surface (we are in $\mathbb R^3$) of the normal vector $\hat n$ of that surface, say $\hat n$ is pointing outward, is zero? In other words, is it ...
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### Why is this vector field not conservative, even though it has a potential? (what is the actual definition of a conservative vector field?)

My question is really ''what is the definition of a conservative vector field''? I've consulted 3 textbooks that all say a vector field $\vec{F}$ is conservative by definition if there exists a ...
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Given $f\colon \mathbb{R}^n\rightarrow \mathbb{R}$ smooth and $\phi \in GL(n)$. What is the Hessian matrix $H_{f\circ \phi} = \left(\frac{\partial ^2 (f\circ \phi)}{\partial x_i\partial x_j}\right)_{... • 155 14 votes 2 answers 3k views ### How would one arrive at the formulas for divergence and curl? It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ... • 26.5k 14 votes 1 answer 2k views ### Observation on rot (curl), div and grad on a vector field? Let$\vec{F}(x, y, z)$be a vector-valued function describing a vector-field. Then the rotation and divergence of the field are:$\nabla \times \vec{F} = \text{curl}(\vec{F}) = \color{red}{(\frac{\...
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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 ...
Consider a smooth vector field $\mathbf u\colon\Omega\to\mathbb R^3$ defined on an open domain $\Omega\subseteq\mathbb R^3$ such that $\mathbf u$ has zero divergence and zero curl on $\Omega$, that is,...