Mathematica Lesson 12: Line Integrals
Mathematica Lesson 12: Line Integrals
In the menu at the top, click Make Your Own Copy to work in Mathematica Online, or click Download to use it in Mathematica.
When asked "Do you want to automatically evaluate all the initialization cells in the notebook...?" I recommend "No" so that you can work through each cell one-at-a-time.
In the menu at the top, click Make Your Own Copy to work in Mathematica Online, or click Download to use it in Mathematica.
When asked "Do you want to automatically evaluate all the initialization cells in the notebook...?" I recommend "No" so that you can work through each cell one-at-a-time.
When asked "Do you want to automatically evaluate all the initialization cells in the notebook...?" I recommend "No" so that you can work through each cell one-at-a-time.
Vector Fields
Vector Fields
Question 1
Question 1
First, let's bring in a command we've seen before. Using Grad in Mathematica, find the gradient vector field for
f(x,y,z)=exp(3x)+y²+ycos(z):
f(x,y,z)=exp(3x)+y²+ycos(z):
Information
Information
Next we look at defining and plotting vector fields of the form F⃗(x,y)=⟨P(x,y),Q(x,y)⟩ or F⃗(x,y,z)=⟨P(x,y,z),Q(x,y,z),R(x,y,z)⟩.
For the rest of the course, I recommend enclosing inputs in curly brackets. This insists that we are plugging vectors into our functions and will make composition (to evaluate line and surface integrals) much easier.
To create the vector field F⃗(x,y)=⟨x²,1−y⟩ in Mathematica, use this:
For the rest of the course, I recommend enclosing inputs in curly brackets. This insists that we are plugging vectors into our functions and will make composition (to evaluate line and surface integrals) much easier.
To create the vector field F⃗(x,y)=⟨x²,1−y⟩ in Mathematica, use this:
In[]:=
F[{x_,y_}]:={x^2,1-y}
(Notice the curly brackets around the inputs x and y.)
You can see this vector field on the domain {(x,y) ∈ ℝ² : −3 ≤ x ≤ 3, −3 ≤ y ≤ 3} using VectorPlot:
You can see this vector field on the domain {(x,y) ∈ ℝ² : −3 ≤ x ≤ 3, −3 ≤ y ≤ 3} using VectorPlot:
In[]:=
VectorPlot[F[{x,y}],{x,-3,3},{y,-3,3}]
It may be easier to get a picture of the "flow" or motion of a vector field using StreamPlot:
In[]:=
StreamPlot[F[{x,y}],{x,-3,3},{y,-3,3}]
Question 2
Question 2
Below, use our new convention to define the vector field F⃗(x,y,z)=⟨x−y,cos(y),z²⟩. Be sure to use Mathematica syntax, and omit any spacing.
Note that you can plot this in 3-D using
VectorPlot3D[F[{x,y,z}],{x,-3,3},{y,-3,3},{z,-3,3}]