Is there any real life question or problem of electrical engineering which can be solved by multi variable calculus or vector calculus vector integration?
[Math] Real life Example of multi variable calculus and vector integration in electrical engineering
integrationmultivariable-calculus
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Multivariable calculus is helpful because it gives many applications of linear algebra, but it's certainly not necessary. In fact, you probably need linear algebra to really start to understand multivariable calculus.
To wit, one of the central objects in multivariable calculus is the differential of a function. In single-variable calculus, you are taught that the differential of a function $f:\mathbb{R}\to\mathbb{R}$ is a new map $f':\mathbb{R}\to\mathbb{R}$ which provides the slope of the tangent line to $f$ at each point in $\mathbb{R}$. This is strictly correct, but it is not the best way to understand single-variable calculus if you want to easily generalize.
The better way to see single-variable calculus is to recall that the tangent line to $f$ at $x$ is the best affine-linear approximation to $f$ at $x$, i.e., $f$ is approximated by $f(y)\approx f'(x)(y - x) + f(x).$
This generalizes quite well! If $f:\mathbb{R}^n\to\mathbb{R}^m$, the differential to $f$ at $x$, $df_x$, is the best linear approximation to $f$ at $x$: $f(y)\approx df_x(y-x) + f(x)$. Now, we think of $x$ and $y$ as vectors in $\mathbb{R}^n$ and the differential $df_x$ is an $n\times m$ matrix.
Even more generally, we think of $df$ as a map from $\mathbb{R}^n$ into $Hom(\mathbb{R}^n,\mathbb{R}^m)$ which measures the best linear approximation of $f$ at each point $x\in\mathbb{R}^n$.
Generalizing further requires the notion, from differential geometry, of a smooth manifold. Such manifolds carry objects called tangent bundles, which assign to each point of the manifold an abstract vector space.
You can see how linear algebra is a little more helpful for multivariable calculus than the other way around.
Good question! Here's a start. The ordinary derivative in one-variable calculus is a Lie derivative along a special vector field on $\mathbb{R}$; in particular, it is not a special case of the exterior derivative. The exterior derivative is instead some kind of "universal derivative": it records all of the information you would need to determine the derivative of a function along any vector field, for example. In particular, unlike the ordinary derivative, the exterior derivative of a function is a different kind of object, namely a $1$-form. Roughly speaking, a $1$-form is "the kind of thing that pairs with a vector field to return a number," so you can see the relationship there to what I said above.
Best Answer
Pretty much every question in electricity and magnetism. Also, most questions in economics, many questions in finance, all of orbital mechanics (how do you think they get rocket ships to go where they're supposed to?).
I guess the short form is "Science." But also finance. Computer graphics (my own bias here). Most of engineering. Some topics in medicine. It's almost easier to answer "Is there a field of endeavor in which these ideas have not been applied?"