Vector Calculus with Python - Gradient, Div, Curl, Stokes, Divergence

Here is how to calculate vector functions in python.

I said I would include links to some other videos- here they are:

2D Green's theorem

Green's theorem visualization

Stoke's Theorem with a more complicated path

Maxwell's equations in differential form

Stoke's Theorem for a hemisphere

Divergence Theorem for a cube (not python)

Numerical methods for div grad curl

[aryan]

You are doing amazing work sirrr.... Please keep it up...

[러블리민희]

if 'I' wanna implement this on Python, you have to use np.array([x, y, z]) not using vector(x, y, z)
The following code is the gradient code on Python:

# Gradient
# g(x, y, z) = y * z^2 * sin(x)
from math import sin
import numpy as np
def g(p):
return p[1] * p[2]**2 * sin(p[0])

def gradg(p):
ds = 0.001
dx = np.array([ds, 0, 0])
dy = np.array([0, ds, 0])
dz = np.array([0, 0, ds])
# dg/dx, dg/dy, dg/dz
# dg/dx = (g(x+Δx) - g(x-Δx)) / 2Δx
# dg/dy = (g(y+Δz) - g(y-Δz)) / 2Δy
# dg/dz = (g(y+Δz) - g(y-Δz)) / 2Δz
# Δx = Δx = Δx = ds = 0.001 --> very small
gx = (g(p + dx) - g(p - dx)) / (2*ds)
gy = (g(p + dy) - g(p - dy)) / (2*ds)
gz = (g(p + dz) - g(p - dz)) / (2*ds)
return np.array([gx, gy, gz])

rt = np.array([1, 1, 1])
print("g(1, 1, ) = ", g(rt))
print("Delg(1, 1, 1) = ", gradg(rt))


# Divergence
def F(p):
return np.array([p[0]**2 * p[1], 2**p[0] * p[2], p[2]**2 - 2*p[1]])

def divF(p):
ds = 0.001
dx = np.array([ds, 0, 0])
dy = np.array([0, ds, 0])
dz = np.array([0, 0, ds])

tempx = (F(p+dx)[0] - F(p-dx)[0]) / (2*ds)
tempy = (F(p+dy)[1] - F(p-dy)[1]) / (2*ds)
tempz = (F(p+dz)[2] - F(p-dz)[2]) / (2*ds)
return tempx + tempy + tempz

rt = np.array([1, 1, 1])
print("F(1, 1, -1) = ", F(rt))
print("DelF(1, 1, 1) = ", divF(rt))

[johnh.g.weisenfeld]

I put something like this in every time you do 3D visualizations to indicate axes.
#put arrows for the axes...as a reference.
axisscale = 0.1
xaxis = arrow(pos=vector(0, 0, 0), shaftwidth=.1*axisscale,
axis=axisscale*vector(1, 0, 0), label="x", color=color.purple)
yaxis = arrow(pos=vector(0, 0, 0), shaftwidth=.1*axisscale,
axis=axisscale*vector(0, 1, 0), label="y", color=color.red)
zaxis = arrow(pos=vector(0, 0, 0), shaftwidth=.1*axisscale,
axis=axisscale*vector(0, 0, 1), label="z", color=color.green)

[HH-mwsq]

Could you please post the full code for the Path Integral, Flux, Stokes theorem, hemisphere, volume integral, and divergence theorem?

[HH-mwsq]

Could you please post the full code for the Path Integral, Flux, Stokes theorem, hemisphere, volume integral, and divergence theorem?

[johnjohn-edqt]

Nice. I have been using your Python/Vpython vids for several years with my students, and this will definitely be added o the list. Side note: nabla was chosen as the formal name for the symbol (typesetting, 'del' being the operator name, in the same sense as octothorpe is the name for symbol '#', though it has many names in application) due to the resemblance to a form of phoenician harp. Yes, I know this due to one of my undergrad profs. Yes, I make sure my student know it. And, yes, I read it as 'del dot F' and 'del cross F'. The same prof was a pedant about the dot and cross product forms for divergence and curl. 'It is not a full-fledged vector! You can not treat it as one!'