Vector Fields

In Vector Calculus, a vector field in Rn\mathbb R^n is a map

F:A⊂Rn→Rn\mathbf F : A\subset\mathbb R^n\to \mathbb R^n 

that assigns to each point in its domain AA

a vector F(x).\mathbf F (\mathbf x).

We can visualize vector fields, whether in the plane or in space, by attaching an arrow to each point.

R2\mathbb R^2

R3\mathbb R^3

R3\mathbb R^3

R2\mathbb R^2

We can visualize vector fields, whether in the plane or in space, by attaching an arrow to each point.

We also just consider a small sample of points to plot the vector field!

We also just consider a small sample of points to plot the vector field!

We also just consider a small sample of points to plot the vector field!

We also just consider a small sample of points to plot the vector field!

Remove the points!

www.geogebra.org/3d

www.geogebra.org/3d

F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle

Multivariable functions

Sequence() command

The variable k

runs from -3 to 3

with an increment of 2

A sequence of numbers!

Use the Zip() command!

Zip(Zip(Zip()))

Expression (a,b,c)

3D array of points in R3\mathbb R^3

a,b,c are variables

that belong to L

Define a scale factor

Apply the map F(x)\mathbf F(\mathbf x)

Apply the map F(x)\mathbf F(\mathbf x)

Apply the map F(x)\mathbf F(\mathbf x)

Multiply by the scale factor:

s⋅F(x)s \cdot\mathbf F(\mathbf x)

LP is a 3D array!

Flatten() command

All vectors start at the origin!

Translate(<Obj>, <Point>)

Fx(x, y, z) = x
Fy(x, y, z) = y
Fz(x, y, z) = z
L = Sequence(k, k, -3, 3, 2)
LP = Flatten( Zip( Zip( Zip( (a, b, c), a, L ), b, L ), c, L ) )
s = 0.5
VF = Zip(Translate(s*(Fx(P), Fy(P), Fz(P)), P), P, LP)

Script 1

Fx(x, y, z) = x
Fy(x, y, z) = y
Fz(x, y, z) = z
L = Sequence(k, k, -3, 3, 2)
LP = Flatten( Zip( Zip( Zip( (a, b, c), a, L ), b, L ), c, L ) )
s = 0.5
LQ = Zip((s * Fx(P), s * Fy(P), s * Fz(P)), P, LP)
VF = Zip(Vector(P, P + Q), P, LP, Q, LQ)

Script 2

www.geogebra.org/calculator

Fx(x, y) = x
Fy(x, y) = y
L = Sequence(k, k, -3, 3, 1)
LP = Flatten( Zip( Zip( (a, b), a, L ), b, L ) )
s = 0.5
VF = Zip( Translate( s*(Fx(P), Fy(P)), P ), P, LP )

Script 1

Fx(x, y) = x
Fy(x, y) = y
L = Sequence(k, k, -3, 3, 1)
LP = Flatten( Zip( Zip( (a, b), a, L ), b, L ) )
s = 0.5
LQ = Zip((s * Fx(P), s * Fy(P)), P, LP)
VF = Zip(Vector(P, P + Q), P, LP, Q, LQ)

Script 2

Vector fields in R2\mathbb R^2

Vector fields in R3\mathbb R^3

Links in the description!

www.desmos.com/3d

www.desmos.com/3d

F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle

List of numbers:

{-3,-2,-1,0,1,2,3}

3D Array of points in R3\mathbb R^3

Apply the map F(x)\mathbf F(\mathbf x)

Define a scale factor

Math3D

www.math3d.org

Remove Explicit Surface

F(x,y,z)=⟨yx2+y2,−xx2+y2,0⟩\mathbf F(x,y,z)= \Biggl\langle \frac{y}{\sqrt{x^2+y^2}}, \frac{-x}{\sqrt{x^2+y^2}}, 0\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle \frac{y}{\sqrt{x^2+y^2}}, \frac{-x}{\sqrt{x^2+y^2}}, 0\Biggl\rangle
F(x,y,z)=⟨yx2+y2,−xx2+y2,0⟩\mathbf F(x,y,z)= \Biggl\langle \frac{y}{\sqrt{x^2+y^2}}, \frac{-x}{\sqrt{x^2+y^2}}, 0\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle \frac{y}{\sqrt{x^2+y^2}}, \frac{-x}{\sqrt{x^2+y^2}}, 0\Biggl\rangle
F(x,y,z)=⟨yx2+y2,−xx2+y2,0⟩\mathbf F(x,y,z)= \Biggl\langle \frac{y}{\sqrt{x^2+y^2}}, \frac{-x}{\sqrt{x^2+y^2}}, 0\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle \frac{y}{\sqrt{x^2+y^2}}, \frac{-x}{\sqrt{x^2+y^2}}, 0\Biggl\rangle
F(x,y,z)=⟨xx2+y2+z2,yx2+y2+z2,zx2+y2+z2⟩\mathbf F(x,y,z)= \Biggl\langle \frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}}\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle \frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}}\Biggl\rangle

Easier to type this!

F(x,y,z)=⟨xx2+y2+z2,yx2+y2+z2,zx2+y2+z2⟩\mathbf F(x,y,z)= \Biggl\langle \frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}}\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle \frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}}\Biggl\rangle

Compare to this!

F(x,y,z)=⟨x,y,z⟩\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle x,y, z\Biggl\rangle
F(x,y,z)=⟨−0.1y,0.1x,0.5z⟩\mathbf F(x,y,z)= \Biggl\langle -0.1y, 0.1x, 0.5z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle -0.1y, 0.1x, 0.5z\Biggl\rangle
F(x,y,z)=⟨−0.1y,0.1x,0.5z⟩\mathbf F(x,y,z)= \Biggl\langle -0.1y, 0.1x, 0.5z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle -0.1y, 0.1x, 0.5z\Biggl\rangle
F(x,y,z)=⟨−0.1y,0.1x,0.5z⟩\mathbf F(x,y,z)= \Biggl\langle -0.1y, 0.1x, 0.5z\Biggl\rangle
\mathbf F(x,y,z)= \Biggl\langle -0.1y, 0.1x, 0.5z\Biggl\rangle

Math3d.org

Christopher Chudzicki

Three.js

Ricardo Cabello (Mr.doob)

Mathbox.js

Steven Wittens

Math3d.org

Christopher Chudzicki

Three.js

Ricardo Cabello (Mr.doob)

Mathbox.js

Steven Wittens

The Code is Open Source

Availabe at GitHub

Link in the description!

Patrons:

Edward Huff, Abei, pmbem, Sophia Wood, Adam Parrott, Doug Kuhlman, Dennis Watson, bleh, Miguel Díaz, Ruan Ramon, Maciej Lasota, Christopher-Alexander Hermanns, Aarón Reyes, Gabriela Sofia Marin Sánchez, Jerome Siegler, Yashar Shoraka, Jeff Butterworth, Scott Pedersen.

Thanks for

watching!

Thanks for

watching!

Patrons:

Edward Huff, Abei, pmbem, Sophia Wood, Adam Parrott, Doug Kuhlman, Dennis Watson, bleh, Miguel Díaz, Ruan Ramon, Maciej Lasota, Christopher-Alexander Hermanns, Aarón Reyes, Gabriela Sofia Marin Sánchez, Jerome Siegler, Yashar Shoraka, Jeff Butterworth, Shaun MacMillan, Scott Pedersen.

Vector fields

By Juan Carlos Ponce Campuzano

Vector fields

Explore the fascinating world of vector fields in Rn, discover engaging visualizations, and access open-source resources to deepen your understanding. https://youtu.be/QB0p8qjL_xU

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