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jverzani 2024-05-25 14:54:27 -04:00
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quarto/integral_vector_calculus/line_integrals.qmd
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@ -316,7 +316,7 @@ $$
This is similar to gravitational force and is a *conservative force*. We saw that a line integral for work in a conservative force depends only on the endpoints. Verify, that for a closed loop the work integral will yield $0$.
Take as a closed loop the unit circle, parameterized by arc-length by $\vec{r}(t) = \langle \cos(t), \sin(t)\rangle$. The unit tangent will be $\hat{T} = \vec{r}'(t) = \langle -\sin(t), \cos(t) \rangle$. The work to move a particle of charge $q_0$ about a partical of charge $q$ at the origin around the unit circle would be computed through:
Take as a closed loop the unit circle, parameterized by arc-length by $\vec{r}(t) = \langle \cos(t), \sin(t)\rangle$. The unit tangent will be $\hat{T} = \vec{r}'(t) = \langle -\sin(t), \cos(t) \rangle$. The work to move a particle of charge $q_0$ about a particle of charge $q$ at the origin around the unit circle would be computed through:
```{julia}
@ -729,7 +729,7 @@ The surface element is the cross product $\langle \cos(\theta), \sin(\theta), -b
```{julia}
@syms 𝑹::postive θ::positive 𝒂::positive 𝒃::positive
@syms 𝑹::positive θ::positive 𝒂::positive 𝒃::positive
𝒏 = [cos(θ), sin(θ), -𝒃] × [-𝑹*sin(θ), 𝑹*cos(θ), 0]
𝒔𝒆 = simplify(norm(𝒏))
```
@ -1213,7 +1213,7 @@ numericq(0)
###### Question
Let $f(x,y) = \tan^{-1}(y/x)$. We will integrate $\nabla{f}$ over the unit circle. The integrand wil be:
Let $f(x,y) = \tan^{-1}(y/x)$. We will integrate $\nabla{f}$ over the unit circle. The integrand will be:
```{julia}
@ -1418,7 +1418,7 @@ numericq(a)
###### Question
Let $F=\langle 0,0,1\rangle$ and $S$ be the upper-half unit sphere, parameterized by $\Phi(\theta, \phi) = \langle \sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta), \cos(\phi)\rangle$. Compute $\iint_S (F\cdot\hat{N}) dS$ numerically. Choose the normal direction so that the answer is postive.
Let $F=\langle 0,0,1\rangle$ and $S$ be the upper-half unit sphere, parameterized by $\Phi(\theta, \phi) = \langle \sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta), \cos(\phi)\rangle$. Compute $\iint_S (F\cdot\hat{N}) dS$ numerically. Choose the normal direction so that the answer is positive.
```{julia}