# Mathematical and Computational Methods in Physics and Engineering (fall 2016)

## Daily outline for Wed, Nov 9, 2016 9:10 AM (1 year ago)

### fundamental theorems of calculus

assigned today:
• standard-fundamental theorems of calculus:

I can explain and use the fundamental theorems of calculus.

1. Compare and contrast the 1D, grad, div, and curl versions of the fundamental theorem of calculus.
2. For $$\vec{f}=(2xz+3y^2)\hat{y}+(4yz^2)\hat{z}$$ check both Stoke's theorem for a single loop with 2 different surfaces and gauss' theorem for an interesting volume.
3. Show that the following are equivalent:
1. $$\vec{\nabla}\times\vec{F}=0$$ everywhere
2. $$\int_\vec{a}^\vec{b}\vec{F}\cdot\vec{dl}$$ is independent of path, for any given end points.
3. $$\oint\vec{F}\cdot\vec{dl}=0$$ for any closed loop.
4. $$\vec{F}$$ is the gradient of some scalar, $$\vec{F}=-\vec{\nabla}V$$.

Due today:

# in class

• talked about the funk we're in after the election
• Quiz on comparing and contrasting the Laplace Transform and Eigen approach to a damped oscillator (with non-zero initial conditions).
• Asked them to write down the fundamental theorem of calculus
• most wrote $$\int_a^b f(x)\,dx=F(b)-F(a)$$
• I walked through the other three and we compared and contrasted (including a one-word story!)
• we talked about when you need a vector field and when you need a scalar field