I am fairly new to calculus (I a learned a bit from the interwebs, not studied it), but I wanted to move to multivariable calculus because while making a 3D rendering program, order of rendering was not quite right, so my guess was to render in order based on the distance. It seems to work, but not as perfectly as I would want, so I tried to use my calculus knowledge to find the closest distance between a triangle and origin or $(0,0,0)$.
My question is, because I dont want to waste 2 hours doing math wrong, when in regular calculus you would note it $$ \frac{dy}{dx} = \lim_{h \to 0} \left(\frac{f(x+h)-f(x)}h\right) $$ then how would you note that properly with say 2 variables? Or am I not supposed do it like this but differently? I was thinking about $$ \frac{dy}{dx+dz} = \lim_{(h_x,h_z)\to(0,0)}\left(\frac{f(x+h_x,z+h_z)-f(x,z)}{h_x+h_z}\right) $$ but I dont know if that makes sense.
I know that you can write $x$ and $z$ as a function of $t$ like $$ \frac{dy}{dt} = \lim_{h\to0}\left(\frac{f(x(t+h), z(t+h)) - f(x(t), z(t))}{h}\right) $$ but this makes it regular calculus and i am not able to make a function that would return all points on a triangle with just a single variable. (I do that by slicing the triangle to infinitely many lines, pick one using $x$ and slice that line into infinitely many points and pick one using $z$. Final goal is to find the minima in interval from $[0,0]$ to $[1,1]$.)
Just to clarify: my question is just about noting this properly, not about the answer to the problem I'm tring to solve with it.
Typically, the in three variable function, you denote the partial derivatives as $\frac{\partial f}{\partial x}= \lim_{h\to 0}\frac{f(x+ h, y, z)- f(x, y, z)}{h}$ $\frac{\partial f}{\partial y}= \lim_{h\to 0}\frac{f(x, y+ h, z)- f(x, y, z)}{h}$ $\frac{\partial f}{\partial z}= \lim_{h\to 0}\frac{f(x, y, z+ h)- f(x, y, z)}{h}$
In general $R^n$ and $R^m$ we can take "the derivative" of a function, f, from $R^n$ to be $R^m$ to be "the linear transformation from $R^n$ to $R^m$ that best approximates f at the given point. In particular, a differentiable function, f(x,y,z), from $R^3$ to R, can be approximated, in a neighborhood of $(x_0, y_0, z_0)$ as $f(x,y,z)= f(x_0,y_0,z_0)+ L(x_0, y_0, z_0)\cdot (x- x_0, y-y_0, z-z_0)+ \Delta(x,y,z)$ where $L(x,y,z)$ is that linear transformation and $\Delta(x,y,z)$ is a function such that $\lim_{(x,y,z)\to (x_0,y_0,z_0)} \frac{\Delta(x,y,z)}{\sqrt{(x-x_0)^2+ (y-y_0)^2+ (z-z_0)^2}}= 0$. It follows that that "linear transformation", $L(x,y,z)$ is the gradient $\nabla f= \left<\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial x}\right>$.
In that sense, we can think of the gradient at each point as being the "derivative" there