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).

Vector analysis, a branch of mathematics that deals with quantities that have both magnitude and direction.

Some physical quantities like the mass or the temperature at some point only have magnitude. We can represent these quantities by number alone (with the appropriate units) and so we call them scalars. There are other physical quantities that have magnitude and direction, called vector. Their magnitude can stretch or shrink, and their direction can reverse. These quantities can be added in such a way that takes into account both direction and magnitude.

Force is an example of a quantity that acts in a certain direction with some magnitude that we measure in Newtons. When two forces act on an object, the sum of the forces depends on both the direction and magnitude of either one. Position, displacement, velocity, acceleration, force, momentum, and torque are all physical quantities that can be mathematically represented by vectors.

One of the most difficult problems in understanding physics is learning how to represent these physical quantities as mathematical vectors.

Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the $~19{th}~$ century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their $~1901~$ book, Vector Analysis.

In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize, as discussed below.

Applications:

When we apply vectors to physical quantities it’s nice to keep in the back of our minds all these formal properties. However from the physicist’s point of view, we are interested in representing physical quantities like displacement, velocity, acceleration, force, impulse, momentum, torque, and angular momentum as vectors. We can’t add force to velocity or subtract momentum from torque. We must always understand the physical context for the vector quantity. So instead of approaching vectors as formal mathematical objects we shall instead consider the following essential properties that enable us to represent physical quantities as vectors.

References:

https://en.wikipedia.org/wiki/Vector_calculus

http://web.mit.edu/8.01t/www/materials/modules/ReviewA.pdf

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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 vector. Reading this definition makes me consider…
Jing
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Gradient of 2-norm squared

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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Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = \int_{\partial M} \omega.$$ Doesn't that look like…
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The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry between the 1-dim version and this version. But as…
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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 found the formula $$\nabla(a\cdot b) = (\nabla a)\cdot…
Euler....IS_ALIVE
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Is there a vector field that is equal to its own curl?

I was wondering if there is a vector field that satisfies the following condition: $$\vec F=\nabla \times \vec F$$
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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 intuition is the idea of slicing the 3D graph of the…
rb612
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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 the usual del operator and $\nabla^2$ is the vector…
Demosthene
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What does the symbol nabla indicate?

First up, this question differs from the other ones on this site as I would like to know the isolated meaning of nabla if that makes sense. Meanwhile, other questions might ask what it means in relation to something else. This might be a very stupid…
Sebastian Nielsen
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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 through scale factors, ie…
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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?
dtg
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Proof of this fairly obscure differentiation trick?

Suppose we're tying to differentiate the function $f(x)=x^x$. Now the textbook method would be to notice that $f(x)=e^{x \log{x}}$ and use the chain rule to find $$f'(x)=(1+\log{x})\ e^{x \log{x}}=(1+\log{x})\ x^x.$$ But suppose that I didn't make…
user1892304
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Why is the gradient always perpendicular to level curves?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. A level set is a set of points: $$L(c) = \{x \in \mathbb{R}^n | f(x) = c\}$$ Two vectors $a, b \in \mathbb{R}^n$ are perpendicular, when their dot product is 0: $$a \perp b :\Leftrightarrow…
Martin Thoma
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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 $A$ that would satisfy this equation? (I know…
Max
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Can a gradient vector field with no equilibria point in every direction?

Suppose that $V:\mathbb{R}^n \to \mathbb{R}$ is a smooth function such that $\nabla V : \mathbb{R}^n \to \mathbb{R}^n$ has no equilibria (i.e. $\forall x \in \mathbb{R}^n : \nabla V (x) \not = 0$). Under these hypotheses, is it possible that $\nabla…
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