I recently learned about tangent planes and the general formulas. This got me wondering if, similar to normal calculus, you could express a multidimensional equation with a taylor series.
You would take the sum from 0 to infinity of (∇^nf(z)(x-z)^n)/n! where as n is the index and z is the list of constant inputs.
Also, I don’t really know if it is OK to put nabla to the power of n.
If your real function of several variables is sufficiently smooth then, yes, you can use directional derivatives in the usual Taylor’s formula:
f(\vec{x}) = \displaystyle\sum_n \frac{\bigl((\vec{x}-\vec{z})\cdot\nabla\bigr)^nf(\vec{z})}{n!}
and also write this in terms of partial derivatives. Here f is a function \mathbb{R}^n \to \mathbb{R}.
The easiest way to see that this works is to consider the vector that passes though z to x, and then consider the values of f along that line. That is a single variable function and so can be approximated with a Taylor series, thus we can estimate its value at one point along the line f(x) in terms of its derivatives on another point on the line f(z).
Explicitly, for any particular vectors z and x to define the function g(a) = f(z+a*(x-z)). This is a one dimensional function in a, and has the property that g(0)=f(z) and g(1)=f(x). Everything else should just fall out from the calculation of the various derivatives dg/da = df(z+a*(x-z))/da as a function of x and z and then plugging then into the standard Taylor formula for estimating g(1) centering on g(0).
Out of curiosity, is there a difference in schools calling the operator nabla vs. del? We always called it del. I know that nabla is the name of the symbol, and is called that in \LaTeX, but I consider del to be the name of the operator. I wonder if this is a British/American thing or the like.
I honestly don’t know, I am learning this during summer vacation on khan academy. On there, they call it both nabla and del. You might be right about del being the operator because they seem to use the term nabla like a symbol whenever they discuss it.
A nabla is what the symbol looks like; compare the delta symbol. These are then used to denote operators, like the gradient or Laplacian (or “del”).
I was always admonished for using “del” (or “del dot” or “del cross”) as the name of the operator, preferring “grad”, “div”, and “curl” (or whatever else) as appropriate.
The explanation given was that the del symbol meant specifically “the vector of derivative operators with respect to each coordinate”, which only matched the operator in question in rectangular Cartesian coordinates.
I’ve only ever heard it called Del. Took Calculus in Oregon.
That’s fair, I guess, and I’d certainly consider those synonyms. But I’d never heard anyone admonished for using del. Maybe it depended on whether one was describing the intent “take the curl of V, then…” vs. reading what’s written “del cross V is equal to…”. It’s also possible there was a difference between pure math vs. physics math or engineering math. The latter two didn’t really need to generalize into >3 dimensions, though they do benefit from non-Cartesian coordinate systems. Curl only works in 3 dimensions, too.
For what it’s worth, among physicists, at most you’ll get one mention “This symbol is technically called nabla”, but the actual word used, for both the symbol and the operator, is always “del”.
Then again, physicists are known to get sloppy about some things, compared to mathematicians.
Another data point: We called both, the operator and the symbol, “nabla” in my German university.
Just by way of example, and expanding at (0,0), if you try to expand f(x,y) as \sum_{n=0,m=0}^{\infty,\infty}a_{n,m}x^ny^m, you will find, assuming this really represents f that a_{n,m}=\frac{1}{n!m!}\frac{\partial^nf}{\partial x^n}\frac{\partial^mf}{\partial y^m}, evaluated at (0,0).
Make the obvious changes evaluated at any other point or in any number of variables