Vector Fields#
Revised
03 Apr 2023
Programming Environment#
Show code cell source
import numpy as np
import pandas as pd
import matplotlib as mpl
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d
from ipywidgets import interactive
plt.style.use('ggplot');
%matplotlib widget
import plotly
import plotly.figure_factory as ff
import plotly.graph_objects as go
from datetime import datetime as d
import locale as l
import platform as p
import sys as s
pad = 20
print(f"{'Executed'.upper():<{pad}}: {d.now()}")
print()
print(f"{'Platform' :<{pad}}: "
f"{p.mac_ver()[0]} | "
f"{p.system()} | "
f"{p.release()} | "
f"{p.machine()}")
print(f"{'' :<{pad}}: {l.getpreferredencoding()}")
print()
print(f"{'Python' :<{pad}}: {s.version}")
print(f"{'' :<{pad}}: {s.version_info}")
print(f"{'' :<{pad}}: {p.python_implementation()}")
print()
print(f"{'Matplotlib' :<{pad}}: {mpl .__version__}")
print(f"{'NumPy' :<{pad}}: {np .__version__}")
print(f"{'Pandas' :<{pad}}: {pd .__version__}")
print(f"{'Plotly' :<{pad}}: {plotly.__version__}")
EXECUTED : 2024-05-21 15:45:59.964261
Platform : 14.4.1 | Darwin | 23.4.0 | arm64
: UTF-8
Python : 3.11.9 | packaged by conda-forge | (main, Apr 19 2024, 18:34:54) [Clang 16.0.6 ]
: sys.version_info(major=3, minor=11, micro=9, releaselevel='final', serial=0)
: CPython
Matplotlib : 3.8.4
NumPy : 1.26.4
Pandas : 2.2.2
Plotly : 5.21.0
Definition of a Vector Field#
Let \(D\subseteq\mathbb{R}^n\).
A vector field on \(\mathbb{R}^n\) is a function \(\mathbf{F}\) that assigns to each point \((x_1,...,x_n)\in D\) an \(n\)-dimensional vector \(\mathbf{F}(x_1,...,x_n)\).
If a point \((x_1,...,x_n)\) is identified with its position vector \(\mathbf{x}=\langle x_1,...,x_n\rangle\), then \(\mathbf{F}\) is a function that assigns to vector \(\mathbf{x}\) an \(n\)-dimensional vector \(\mathbf{F(x)}\).
\(\mathbb{R}^2\)
\( \mathbf{F}(x,y) =P(x,y)\mathbf{i}+Q(x,y)\mathbf{j} =\langle P(x,y),Q(x,y)\rangle \)
where the component functions \(P\) and \(Q\) are scalar fields.
\(\mathbb{R}^3\)
\( \mathbf{F}(x,y,z) =P(x,y,z)\mathbf{i}+Q(x,y,z)\mathbf{j}+R(x,y,z)\mathbf{k} =\langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle \)
where the component functions \(P\), \(Q\), and \(R\) are scalar fields.
Continuity of Vector Fields
\(\mathbf{F}\) is continuous iff its component functions are continuous.
Examples of Vector Fields#
Examples of vector fields
Velocity fields
Fluid flows
Force fields
Electromagnetic fields
Gravitational fields
Vortices
Gradient fields
Example of a Vortex
\( \boxed{\mathbf{F(x)}=\langle-y,x\rangle} \)
\( \begin{aligned} \mathbf{F}(x,y) &=\langle f_1(x,y),f_2(x,y)\rangle &&=f_1(x,y)\mathbf{i}+f_1(x,y)\mathbf{j} \\ &=\langle-y,x\rangle &&=-y\mathbf{i}+x\mathbf{j} \end{aligned} \)
Some representative vectors.
\( \begin{aligned} \mathbf{F}(0,0)&=\langle0,0\rangle \\ \mathbf{F}(1,0)&=\langle0,1\rangle \\ \mathbf{F}(0,1)&=\langle-1,0\rangle \\ \mathbf{F}(-1,0)&=\langle0,-1\rangle \\ \mathbf{F}(0,-1)&=\langle1,0\rangle \\ \mathbf{F}(1,1)&=\langle-1,1\rangle \\ \mathbf{F}(-1,1)&=\langle-1,-1\rangle \\ \mathbf{F}(-1,-1)&=\langle1,-1\rangle \\ \mathbf{F}(1,-1)&=\langle-1,-1\rangle \end{aligned} \)
Note
\( \begin{aligned} \mathbf{x\cdot F(x)} =\langle x,y\rangle\cdot\langle-y,x\rangle =-xy+xy =0 \end{aligned} \)
\(\therefore\mathbf{F(x)}\perp\mathbf{x}\) and so \(\mathbf{F(x)}\) is tangent to a circle with center the origin and radius \(|\mathbf{x}|=\sqrt{x^2+y^2}\)
Also note
\( |\mathbf{F(x)}| =\sqrt{(-y)^2+x^2} =\sqrt{x^2+y^2} =|\mathbf{x}| \)
Therefore, the magnitude of the vector \(\mathbf{F(x)}\) is equal to the radius of the circle.
Show code cell source
c=4
n =np.linspace(-c,c,2*c+1)
x,y=np.meshgrid(n,n)
u1 =-y
v1 = x
M =np.hypot(u1,v1)
fig = plt.figure(figsize=(10,5));
ax1 = plt.subplot(121);
ax1.set_aspect(1);
ax1.quiver(x,y,u1,v1,M,pivot='tail',scale=30);
ax1.scatter(x,y,c='red',s=2.7);
ax1.set_xticks(np.arange(-c,c+1,1));
ax1.set_yticks(np.arange(-c,c+1,1));
ax1.set_xlim(-c,c);
ax1.set_ylim(-c,c);
ax1.set_title('$\mathbf{F}(\mathbf{x})=\\langle-y,x\\rangle$');
u2 = y
v2 =-x
M =np.hypot(u2,v2)
ax2 = plt.subplot(122);
ax2.set_aspect(1);
ax2.quiver(x,y,u2,v2,M,pivot='tail',scale=30);
ax2.scatter(x,y,c='red',s=2.7);
ax2.set_xticks(np.arange(-c,c+1,1));
ax2.set_yticks(np.arange(-c,c+1,1));
ax2.set_xlim(-c,c);
ax2.set_ylim(-c,c);
ax2.set_title('$-\mathbf{F}(\mathbf{x})=\\langle y,-x\\rangle$');
Show code cell source
fig = ff.create_quiver(x,y,u1,v1,scaleratio=1)
fig.update_layout(height=500,width=500,margin=dict(l=10,r=10,t=10,b=10))
\( \begin{aligned} \mathbf{F(x)}&=\langle y,\sin x\rangle \end{aligned} \)
Show code cell source
c=6
n =np.linspace(-c,c,2*c+1)
x,y=np.meshgrid(n,n)
u1 =y
v1 =np.sin(x)
M =np.hypot(u1,v1)
fig = plt.figure(figsize=(10,5));
ax1 = plt.subplot(121);
ax1.set_aspect(1);
ax1.quiver(x,y,u1,v1,M,scale=50);
ax1.scatter(x,y,c='red',s=2.7);
ax1.set_xticks(np.arange(-c,c+1,1));
ax1.set_yticks(np.arange(-c,c+1,1));
ax1.set_xlim(-c,c);
ax1.set_ylim(-c,c);
ax1.set_title('$\mathbf{F}(\mathbf{x})=\\langle y,\\sin x\\rangle$');
u2 =-y
v2 =-np.sin(x)
M =np.hypot(u2,v2)
ax2 = plt.subplot(122);
ax2.set_aspect(1);
ax2.quiver(x,y,u2,v2,M,scale=50);
ax2.scatter(x,y,c='red',s=2.7);
ax2.set_xticks(np.arange(-c,c+1,1));
ax2.set_yticks(np.arange(-c,c+1,1));
ax2.set_xlim(-c,c);
ax2.set_ylim(-c,c);
ax2.set_title('$-\mathbf{F}(\mathbf{x})=\\langle-y,-\\sin x\\rangle$');
Show code cell source
fig = ff.create_quiver(x,y,u1,v1,scaleratio=1)
fig.update_layout(height=500,width=500,margin=dict(l=10,r=10,t=10,b=10))
\( \mathbf{F(x)}=\langle\ln(1+y^2),\ln(1+x^2)\rangle \)
Show code cell source
c=6
n =np.linspace(-c,c,2*c+1)
x,y=np.meshgrid(n,n)
u1 =np.log(1+y**2)
v1 =np.log(1+x**2)
M =np.hypot(u1,v1)
fig = plt.figure(figsize=(10,5));
ax1 = plt.subplot(121);
ax1.set_aspect(1);
ax1.quiver(x,y,u1,v1,M,scale=50);
ax1.scatter(x,y,c='red',s=2.7);
ax1.set_xticks(np.arange(-c,c+1,1));
ax1.set_yticks(np.arange(-c,c+1,1));
ax1.set_xlim(-c,c);
ax1.set_ylim(-c,c);
ax1.set_title('$\mathbf{F}(\mathbf{x})=\\langle\\ln(1+y^2),\\ln(1+x^2)\\rangle$');
u2 =-np.log(1+y**2)
v2 =-np.log(1+x**2)
M =np.hypot(u2,v2)
ax2 = plt.subplot(122);
ax2.set_aspect(1);
ax2.quiver(x,y,u2,v2,M,scale=50);
ax2.scatter(x,y,c='red',s=2.7);
ax2.set_xticks(np.arange(-c,c+1,1));
ax2.set_yticks(np.arange(-c,c+1,1));
ax2.set_xlim(-c,c);
ax2.set_ylim(-c,c);
ax2.set_title('$-\mathbf{F}(\mathbf{x})=\\langle-\\ln(1+y^2),-\\ln(1+x^2)\\rangle$');
Show code cell source
fig = ff.create_quiver(x,y,u1,v1,scaleratio=1)
fig.update_layout(height=500,width=500,margin=dict(l=10,r=10,t=10,b=10))
\( \mathbf{F(x)}=z\mathbf{k}=\langle 0,0,z\rangle \)
Show code cell source
c=6
n =np.arange(-c,c+1,3)
x,y,z=np.meshgrid(n,n,np.arange(-2*c,2*c+1,2))
u=0*x
v=0*y
w=z
fig = plt.figure();
ax = plt.subplot(projection='3d');
ax.quiver(x,y,z,u,v,w,length=0.2);
ax.scatter(x,y,z,c='blue',s=2.7);
ax.set_xlabel('x');
ax.set_ylabel('y');
ax.set_zlabel('z');
ax.view_init(10,20);
Show code cell source
fig = go.Figure(data=go.Cone(
x=x.flatten(),
y=y.flatten(),
z=z.flatten(),
u=u.flatten(),
v=v.flatten(),
w=w.flatten(),
colorscale='Blues',
sizemode='absolute',
sizeref=10,
))
fig.update_layout(
height=600,width=600,
margin=dict(l=10,r=10,t=10,b=10),
scene=dict(
aspectratio=dict(x=1,y=1,z=1),
camera_eye=dict(x=1.75,y=1.5,z=0.25)
)
)
\( \mathbf{F(x)}=\langle y,z,x\rangle \)
Show code cell source
c=1
s=0.3
n =np.arange(-c,c+s,s)
x,y,z=np.meshgrid(n,n,np.arange(-c,c+s,s))
u=y
v=z
w=x
fig = plt.figure();
ax = plt.subplot(projection='3d');
ax.quiver(x,y,z,u,v,w,length=0.2);
ax.scatter(x,y,z,c='blue',s=2.7);
ax.set_xlabel('x');
ax.set_ylabel('y');
ax.set_zlabel('z');
ax.view_init(55,75);
Show code cell source
fig = go.Figure(data=go.Cone(
x=x.flatten(),
y=y.flatten(),
z=z.flatten(),
u=u.flatten(),
v=v.flatten(),
w=w.flatten(),
colorscale='Blues',
sizemode='absolute',
))
fig.update_layout(
height=600,width=600,
margin=dict(l=10,r=10,t=10,b=10),
scene=dict(
aspectratio=dict(x=1,y=1,z=1),
camera_eye=dict(x=2,y=1,z=2)
)
)
\( \mathbf{F(x)}=\langle y,-2,x\rangle \)
Show code cell source
c=1
s=0.4
n =np.arange(-c,c+s,s)
x,y,z=np.meshgrid(n,n,np.arange(-c,c+s,s))
u=y
v=-2+0*x
w=x
fig = plt.figure();
ax = plt.subplot(projection='3d');
ax.quiver(x,y,z,u,v,w,length=0.2);
ax.scatter(x,y,z,c='blue',s=2.7);
ax.set_xlabel('x');
ax.set_ylabel('y');
ax.set_zlabel('z');
ax.view_init(20,30);
Show code cell source
fig = go.Figure(data=go.Cone(
x=x.flatten(),
y=y.flatten(),
z=z.flatten(),
u=u.flatten(),
v=v.flatten(),
w=w.flatten(),
colorscale='Blues',
sizemode='absolute',
))
fig.update_layout(
height=600,width=600,
margin=dict(l=10,r=10,t=10,b=10),
scene=dict(
aspectratio=dict(x=1,y=1,z=1),
camera_eye=dict(x=2,y=1,z=1)
)
)
\( \begin{aligned} \mathbf{F(x)}=\left\langle\frac{y}{z},-\frac{x}{z},\frac{2}{4}\right\rangle \end{aligned} \)
Show code cell source
c=1
s=0.4
n =np.arange(-c,c+s,s)
x,y,z=np.meshgrid(n,n,np.arange(-c,c+s,s))
u= y/z
v=-x/z
w= z/4
fig = plt.figure();
ax = plt.subplot(projection='3d');
ax.quiver(x,y,z,u,v,w,length=0.05);
ax.scatter(x,y,z,c='blue',s=2.7);
ax.set_xlabel('x');
ax.set_ylabel('y');
ax.set_zlabel('z');
ax.view_init(25,40);
Show code cell source
fig = go.Figure(data=go.Cone(
x=x.flatten(),
y=y.flatten(),
z=z.flatten(),
u=u.flatten(),
v=v.flatten(),
w=w.flatten(),
colorscale='Blues',
sizemode='absolute',
))
fig.update_layout(
height=600,width=600,
margin=dict(l=10,r=10,t=10,b=10),
scene=dict(
aspectratio=dict(x=1,y=1,z=1),
camera_eye=dict(x=2,y=1,z=1)
)
)
Show code cell source
c=6
n =np.arange(-c,c+1,0.1)
x,y=np.meshgrid(n,n)
u1 =(x+1)/((x+1)**2+y**2)-(x-1)/((x-1)**2+y**2)
v1 =y/((x+1)**2+y**2)-y/((x-1)**2+y**2)
fig = plt.figure(figsize=(10,5));
ax1 = plt.subplot(121);
ax1.set_aspect(1);
ax1.streamplot(x,y,u1,v1,density=1.5,linewidth=0.5,color='#A23BEC');
ax1.plot(-1,0,'-or');
ax1.plot( 1,0,'-og');
ax1.set_xticks(np.arange(-c,c+1,1));
ax1.set_yticks(np.arange(-c,c+1,1));
ax1.set_xlim(-c,c);
ax1.set_ylim(-c,c);
u2 =-((x+1)/((x+1)**2+y**2)-(x-1)/((x-1)**2+y**2))
v2 =-(y/((x+1)**2+y**2)-y/((x-1)**2+y**2))
ax2 = plt.subplot(122);
ax2.set_aspect(1);
ax2.streamplot(x,y,u2,v2,density=1.5,linewidth=0.5,color='#A23BEC');
ax2.plot(-1,0,'-or');
ax2.plot( 1,0,'-og');
ax2.set_xticks(np.arange(-c,c+1,1));
ax2.set_yticks(np.arange(-c,c+1,1));
ax2.set_xlim(-c,c);
ax2.set_ylim(-c,c);
fig.suptitle('Electromagnetic Field');