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“We consider Area as a vector.”

How is an area a vector? Why is that the vector is always normal to the area element?

- Why does dividing a vector by its norm give a unit vector?
- mathematical difference between column vectors and row vectors
- Proving the triangle inequality for the $l_2$ norm $\|x\|_2 = \sqrt{x_1^2+x_2^2+\cdots+x_n^2}$
- Levi civita and kronecker delta properties?
- Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Intuition?
- The proof of the Helmholtz decomposition theorem through Neumann boundary value problem

- Lie vs. covariant derivative: Visual motivation
- Does the Divergence Theorem Work on a Surface?
- Find plane by normal and instance point + distance between origin and plane
- Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Intuition?
- How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?
- Derivative of cross-product of two vectors
- real meaning of divergence and its mathematical intuition
- Why isn't the directional derivative generally scaled down to the unit vector?
- Divergence of $\vec{F} = \frac{\hat{\mathrm{r}}}{r^{2}}$
- Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$

It’s just a definition that is mostly useful in physics. If you want to integrate the flow of a liquid or an electromagnetic field out of a given surface (say the surface of a ball), you would like to multiply each differential surface area by the corresponding flow. But what if the flow isn’t perpendicular to the surface? in this case we want to multiply the flow (which is a vector) with the surface normal.

So instead of writing:

$${\bf{v}}\cdot{\bf{n}}\ dA$$

We can now write:

$${\bf{v}}\cdot{\bf{dA}}$$

It’s a cheat, a mathematical sleight of hand. In 3d, a planar subspace will always have a unique normal direction, and so you can get away with using those normal vectors instead of introducing a lot of extra mathematical framework. Using normal vectors makes sense because these vectors are unique up to a sign and magnitude.

In spaces with more than three dimensions, planes no longer have normal vectors, and dealing with other frameworks to handle such objects is unavoidable.

**Edit**: using clifford algebra, you can directly handle areas (planes) as *bivectors*. The product of vectors that produces such objects is not the cross product but the *wedge* product. The wedge product is anticommutative, so $a \wedge b = – b \wedge a$, similar to the cross product, and in addition $a \wedge a = 0$.

Here’s an example. Let $u = 3\hat x + 2 \hat y$ and $v = 5 \hat y + 2 \hat z$. The area of a parallelogram spanned by these two vectors is

$$A = u \wedge v = (3 \hat x + 2 \hat y) \wedge (5 \hat y + 2 \hat z) = 15 \, \hat x \wedge \hat y + 6 \, \hat x \wedge \hat z + 4 \, \hat y \wedge \hat z$$

Compare this with the corresponding calculation using the cross product. The components are all the same, but the interpretation of the object is different. Again, the main reason for doing this instead of using pseudovectors is that this approach is valid in any number of dimensions. In addition, some aspects of linear transformations are not obvious using pseudovectors (why should a pseudovector transform differently than the vectors used to build it?) but can be seen more intuitively using bivectors. One such case is inversion through the origin.

Actually, you can have a n-dimensional cross-product

which generalizes the 3-dimensional cross-product.

If $A_i, i=1, n-1$

are $n-1$ $n$-dimensional vectors

with the elements of $A_i$

being

$a_{i,1}, a_{1,2}, …, a_{i, n}$,

and the $n$ unit vectors are

$e_1, …, e_n$,

then the following determinant

is a n-dimensional vector orthogonal to

all the $A_i$:

$\left|

\begin{array}{cccc}

e_1 & e_2 & … & e_n \\

A_{1,1} & A_{1,2} & … & A_{1,n} \\

A_{2,1} & A_{2,2} & … & A_{2,n} \\

… & … & … & … \\

A_{k,1} & A_{k,2} & … & A_{k,n} \\

… & … & … & … \\

A_{n-1,1} & A_{n-1,2} & … & A_{n-1,n} \\

\end{array} \right|$

- Problems that become easier in a more general form
- Commutation when minimal and characteristic polynomial agree
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- Need help with the integral $\int_{0}^\infty e^{-x^{2}}x^{2n+1}dx $
- Prove that a polytope is closed
- Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$
- What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve
- Algebra of Random Variables?
- $\int_0^{\pi}{x \over{a^2\cos^2 x+b^2\sin^2 x}}dx$
- Creating a sequence that does not have an increasing or a decreasing sequence of length 3 from a set with 5 elements
- Lipschitz maps between Riemannian manifolds
- Geometry with Trigonometry
- Uniform thickness border around skewed ellipse?
- If $p$ is an odd prime, does every Sylow $p$-subgroup contain an element not in any other Sylow $p$-subgroup?
- Spectral Measures: Reducibility