daviddoji/CalculusWithJuliaNotes.jl
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use quarto, not Pluto to render pages

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jverzani
2022-07-24 16:38:24 -04:00
parent 93c993206a
commit 7b37ca828c
879 changed files with 793311 additions and 2678 deletions
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CwJ/differentiable_vector_calculus/vector_fields.jmd
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@@ -170,15 +170,12 @@ zs = [Z(theta, phi) for theta in thetas, phi in phis]
surface(xs, ys, zs) ## see note
```
```julis; echo=false
note("""Only *some* backends for `Plots` will produce this type of plot. Both `plotly()` and `pyplot()` will, but not `gr()`.
""")
```
!!! note
Only *some* backends for `Plots` will produce this type of plot. Both `plotly()` and `pyplot()` will, but not `gr()`.
```julia; echo=false
note("""Note: PyPlot can be used directly to make these surface plots: `import PyPlot; PyPlot.plot_surface(xs,ys,zs)`""")
```
!!! note
PyPlot can be used directly to make these surface plots: `import PyPlot; PyPlot.plot_surface(xs,ys,zs).
Instead of the comprehension, broadcasting can be used
@@ -856,8 +853,8 @@ choices = [
"Yes",
"No, it is the transpose"
]
ans=2
radioq(choices, ans, keep_order=true)
answ=2
radioq(choices, answ, keep_order=true)
```
###### Question
@@ -879,8 +876,8 @@ choices = [
"Yes",
"No, it is the transpose"
]
ans=1
radioq(choices, ans, keep_order=true)
answ=1
radioq(choices, answ, keep_order=true)
```
###### Question
@@ -907,8 +904,8 @@ choices = [
"The determinant of the Hessian.",
"The determinant of the gradient."
]
ans = 1
radioq(choices, ans, keep_order=true)
answ = 1
radioq(choices, answ, keep_order=true)
```
@@ -920,8 +917,8 @@ choices = [
"`det(hessian(F(lambda, phi), [lambda, phi]))`",
"`det(gradient(F(lambda, phi), [lambda, phi]))`"
]
ans=1
radioq(choices, ans, keep_order=true)
answ=1
radioq(choices, answ, keep_order=true)
```
###### Question
@@ -934,8 +931,8 @@ choices = [
raw"`` 2x^3y/ (z\cos(z) + \sin(z) + 1)``",
raw"`` 3x^2y^2``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
@@ -949,8 +946,8 @@ choices = [
raw"`` \frac{x \left(2 x^{2} - y^{2} z^{2}{\left (x,y \right )}\right)}{\left(x^{2} y^{2} - 2 z^{2}{\left (x,y \right )}\right) z{\left (x,y \right )}}``",
raw"`` \frac{x \left(2 x^{2} - z^{2}{\left (x,y \right )}\right)}{\left(x^{2} - 2 z^{2}{\left (x,y \right )}\right) z{\left (x,y \right )}}``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -965,8 +962,8 @@ choices = [
raw"`` S/r``",
raw"`` R``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
Let $\phi = r^k$. What is $\nabla{\phi}$?
@@ -977,8 +974,8 @@ choices = [
raw"`` kr^k R``",
raw"`` k r^{k-2} S``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
Based on your last answer, are all radial fields $R/r^n$, $n\geq 0$ gradients of scalar functions?
@@ -995,8 +992,8 @@ choices = [
raw"`` S/r``",
raw"`` S``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
Express $S/r^n = \langle F_x, F_y\rangle$. For which $n$ is $\partial{F_y}/\partial{x} - \partial{F_x}/\partial{y} = 0$?
@@ -1008,8 +1005,8 @@ L"As the left-hand side becomes $(-n+2)r^{-n}$, only $n=2$.",
L"All $n \geq 0$",
L"No values of $n$"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
(The latter is of interest, as only when the expression is $0$ will the vector field be the gradient of a scalar function.)