A Trinity of Theorems

There are three essential theorems that capture the fundamentals of multivariable calculus. (And no, I'm not referring to the fundamental theorem of calculus. That's far to simple!)

1. Green's Theorem
Green's theorem essentially says that you can take any positively oriented closed curve in the plane and rewrite it as a double integral over the area enclosed by the curve. In other words:

2. Divergence Theorem
The divergence theorem says that you can take any non-closed surface, S, in space with a boundary, C, and describe the flux across the surface as a triple integral of the divergence in the volume enclosed by the surface.

3. Stokes' Theorem
And to wrap everything together, Stokes' Theorem says that a loop integral on the boundary, C, of a non-closed surface, S, can be rewritten as a double integral of the curl of a vector field, F, across the surface.

These three theorems can be said to be "fundamental" because they relate the boundaries of surfaces to the interior of those surfaces, just as in basic calculus, the "interior" of a one-dimensional curve in the plane can be related to the endpoints of that curve.

Parametric Surfaces

How do you turn a one-dimensional vector-valued function r(t) into two-dimensional surface? Rhetorically, that's like asking how to turn f(x) into f(x,y)--you just add another variable. But instead of making the vector function r(t,z), the conventional form is r(u,v).

For instance, take <cos(u), sin(u), v>, where u ranges from 0 to 2pi and v ranges from 0 to 3. The x- and y-components dictate that in the xy-plane, you have a circle. And what does that circle do? It gets extruded upwards to 3. The end result is a cylinder with radius 1 and height 3.

What is the purpose of having another way to describe a surface? The primary reason is because the function is easier to interpret.

In the 2D plane, we're familiar with:

as a circle of radius 3. But in 3-space, that becomes a cylinder. But what's to distinguish one from another? In parametric, however, this difference is clear. 

Hyperbolic Trig Functions

And just when you thought that you were rid of them for the rest of your math career, they're back (with a vengeance)! Well...not quite. We should all be familiar with the family of trigonometric functions: sine, cosine, tangent, secant, co-secant, and co-tangent. And we shouldn't forget all of their lovely derivatives. And the derivatives of their inverse functions. Well, don't fret when I tell you that there's more.

Hyperbolic trig functions are a separate class of functions that look like sines and cosines, and similarly, also have somewhat familiar derivatives and other properties. For instance, take sinh (x), the hyperbolic sine

function, pronounced "sin-sh". Though it is defined as:

 it's derivative, like sin (x) is just what you'd think: cosh (x), the hyperbolic cosine, pronounced "cosh". But what's interesting is that the derivative of cosh (x), defined as:

isn't -sinh (x), like how the derivative of cos(x) is -sin(x). Rather, it's just sinh(x). So the sinh(x) and cosh(x) have cyclic derivatives!

Some other "look alike" derivatives of hyperbolic trig functions include:

As of yet, I haven't noticed any pattern to help memorize which derivatives are "mirror-images" of each other, but if anyone does discover something, drop a comment!

Line Integrals

Line integrals calculate summations of some quantifiable object X (for instance, density) over a path. We discussed in class how line integrals can also apply to paths that turn and go around in space. To calculate such integrals, we simply need to describe the path as a piece-wise function and integrate accordingly.

In one of our problems, we reversed the path our particle in our line integral would take and we got the exact opposite answer. But what if instead of reversing the path, we reversed the vector field? The answer we arrived at was that it would not yield the same answer. The reversed vector field wouldn't affect the path that the particle took in the same way.

If you look closely, the angle between the original vector field and the tangent to the curve is slightly larger than that of the reversed vector field. So reversing a vector field won't necessarily generate the opposite amount of net work done by the particle, but does yield some interesting phenomenon in physics.

For instance, if you shatter a glass, every resulting shard has a force vector pointing outwards from the origin of, shall we say, collapse? Well, what if we reversed all those vectors? Would the glass come back together? (Link below)

Skip to around 6 seconds and imagine freezing a glass that had just shattered and reversing all those force vectors. And perhaps this would happen.

It's definitely an interesting thing to picture in your head!

Vector Fields - Conservative Fields in Space

As it turns out, determining whether or not a vector field is conservative in space is radically different than that of a vector field in the plane. To determine whether or not a vector field is conservative in space (and by this, I mean 3D vector fields), we need to take the cross product of the gradient and our vector function. This yields a matrix:

This is called the curl of vector field F. If the curl is the zero vector, then the vector field is conservative.