Vector Calculus for N00bs?

Question

Hi. I just finished calculus, and I want to learn vector calculus. So today, I read some stuff on del operators and partial derivatives and I think I understand that ...but I don't know ...
Can somebody PLEASE just explain gradient operators in extremely simple terms that someone with just basic calculus can understand. ..I don't need to have an in depth understanding, I just want to be able to have fun with it because I'm bored right now and that sounds super sweet ....like try to do basic work/energy problems using it ...but I'm not sure exactly how it works ...

Can anybody spare me the technical stuff and show me how to do it in basic terms??

thaks

Answer 1

It should be in your calculus book. It is *sweet*, but I am out of practice and dare not embarass myself by saying something ignorant.

Partial derivatives are easy, if you relax. A nightmare if you treat them as more complicated than they are.

The del operator is pretty much taking the gradient.
Think of a car windshield or a hill with compound curves. Think of the dimensions x,y,z and time. Your gradient might be the path air will take across the side of the windshield or the gully on the hill.
The dimensions can be chemical concentration, such as CO2. A mosquito's brain is finding the gradient numerically (analog!!) A trail of blood in the ocean and a shark homing in on it. The gradient will tell you the ideal path the shark will swim to the victim. The shock wave ripping through high explosive. How will it propagate? The answer is to the right of the del!!

Answer 2

OK, you asked. Start with implicit differentiation. Say we have some function, y, that depends on x. e.g.:

y=5x^3

Apply the differential operator, d, on both sides. Recall from calculus that d(f(g(x)) = f'(x)*g'(x). . .

d(y) = d(5x^3)

dy = (15x^2)dx . . . or dy/dx = 25x^2

Now imagine you have more than one variable. Then the change "d" of something depend on the changes of the thing it depends on. This means you must take several derivatives, one with respect to each variable following the rules of calculus. For example:

z(x,y) = xy^2

dz = y^2(dx) + 2xy(dy)

This gives the change in z given the changes in x and y. The gradient is the vector given by the derivitives in front of the infenitesmals above. The gradient thus has one element for every variable of the thing you're taking the gradient depends on. These individual elements are called "partial derivatives" a derivative taken with respect only one of the many variables. This is what dell means. To actually get profecient with what this stuff can do, you're going to need to find a math book and practice.