# Multivariable Calculus Mathematica commands

Here are the commands I find most useful for MA 242: Multivariable Calculus.

## General Algebra and Calculus |
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The following items are useful throughout the course.

##### Using mathematical functions

Existing functions in Mathematica are capitalized and use square brackets. Example:

Cos[x]

Sin[Pi/3]

Log[7] % this is natural log

Exp[.4*x] % this is e^(.4x)

##### Simplifying Answers

Try appending //Simplify or //FullSimplify at the end of any answer. Example:

Cos[x]^2 + Sin[x]^2 // FullSimplify

##### Avoiding hyperbolic trig functions

Try appending //TrigToExp after an answer. Example:

ArcTanh[u^2] // TrigToExp

##### Returning answers as a decimal:

Append //N at the end of any answer. Example:

Sqrt[3]/17 // N

##### Solving Equations

Solve an equation in Mathematica.

Solve[x^2==9,x]

Solve[x^2-2*x-6==0,x]

Getting answers with the imaginary number Mathematica allows for complex numbers. Remove them by insisting the variables are real:

Solve[2212 == 1467*Exp[13*t], t!Element] should get converted to the "is an element of" symbol .

Example: find the general form for the plane

Solve[{3,2,6}.({x,y,z}-{-1,3,2})==0,z]This will return the plane in the form Rearrange terms to get the general form.

You can also solve a system of equations, e.g.

Solve[3*x+4*y==7 && 2*x-y==-3,{x, y}]

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##### Single Variable Calculus

This is how you define a function of one variable (used for curves in Unit 2):

f[x_]:=x^2+Sin[x]

How to differentiate:

f'[x]

or

D[f[x],x]

(The latter is more similar to partial derivatives in Unit 3.)

To compute the derivative at a certain value, like use "slash-dot:"

f'[x]/.x->4

How to compute the indefinite integral (does not include ):

Integrate[f[x],x]

Definite integral

Integrate[f[x],{x,0,5}]

## Unit 1. Vector Computations |
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We will use need dot products, cross products, and vector magnitudes in many computations throughout the course (especially in Unit 6 and Unit 7), so I hope these commands in particular assist you.

##### Dot Product

Calculate the dot product (inner product) of two vectors.

Dot[{a1,b1,c1},{a2,b2,c2}]

u.v

**Example:**

Dot[{2,3,1},{1,-1,4}]% Computes the dot product of {2, 3, 1} and {1, -1, 4}

{2,3,1}.{1,-1,4}% Computes the dot product of {2, 3, 1} and {1, -1, 4}

##### Cross Product

Save yourself several minutes and compute the cross product of two vectors--I recommend doing this in parallel to your written work so that you can check yourself as you move through problems.

Cross[{a1,b1,c1},{a2,b2,c2}]

**Example:**

Cross[{2,3,1},{1,-1,4}]% Computes the cross product of {2, 3, 1} and {1, -1, 4}

##### Vector Magnitude

Since we can calculate the magnitude (length) of a vector using `Sqrt`

and `Dot`

.

Sqrt[v.v]

**Example:**

u={3,4,5}

Sqrt[u.u]% Computes the magnitude of vector {3, 4, 5} using the dot product

##### Angle between vectors

This is a combination of dot and magnitude, with ArcCos. The answer will be in radians:

ArcCos[u.v/Sqrt[u.u*v.v]]

To get the angle in degrees, just convert this from radians to degrees...

ArcCos[u.v/Sqrt[u.u*v.v]]*180/Pi

##### Projection of onto

There is a command for this, but why bother when it can be done using the definition of projection?

(u.v)/(v.v)*v

**Example:**

u={3,4,5}; v={1,1,1}

(u.v)/(v.v)*v% Computes the projection of {3, 4, 5} onto {1, 1, 1}

## Unit 2. Parametric Curves |
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Be sure to see the page **Curve Quantities**. In many of these computations, it may be useful to append **//FullSimplify** onto the end of your command.

##### Define a curve in parametric form:

r[t_]:={x[t],y[t],z[t]}

Example:

r[t_]:={2*Cos[t],2*Sin[t],t}

##### Plotting a curve in 3D-space:

ParametricPlot3D[r[t],{t,0,6*Pi}]

##### Computing velocity, speed, and acceleration:

r'[t] Sqrt[r'[t].r'[t]] r''[t]

This is just a combination of earlier commands based on the literal definition of

T[t_]:=r'[t]/Sqrt[r'[t].r'[t]]

##### Arc Length

Given that this computation is another combination of earlier commands on this page. Here is an example computing the length of a parametric curve from to

Integrate[Sqrt[r'[t].r'[t]],{t,0,6}]

Figuring out how to use prior commands (like to create ) to compute the curvature is Exercise 5 in **Mathematica Lesson 6 - Curves 2**.

Again, a combination of prior commands based on the definition

T'[t]/Sqrt[T'[t].T'[t]]

Again, a combination of prior commands based on the definition

Cross[T[t],N[t]]

## Unit 3. Differentiable Multivariable Calculus |
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##### Creating a Function of Multiple Variables

To define a function of multiple variables in Mathematica, use the following syntax:

f[x_,y_]:=x^2+y^2

g[x_,y_,z_]:=Sin[x]+Cos[y]-Exp[z]

##### Graphing

Plot3D is used to plot a function of two variables in a 3D space. In the example below, a previously defined function is being plotted on the domain

Plot3D[f[x y],{x,-2,2},{y,-3,3}]

##### Creating contour diagrams

ContourPlot can be used to visualize level curves for a function of two variables. In the example below, the level curves for a previously defined function are

being plotted on the domain

ContourPlot[f[x,y],{x,-2,2},{y,-3,3}]

##### Plotting parametric surfaces

ParametricPlot3D can be used to plot a parametric surface. Below is how to plot the surface parametrized as

ParametricPlot3D[{Sin[u],Cos[v],u+v},{u,0,2*Pi},{v,0,2*Pi}]

##### Partial Derivatives

Partial derivatives of a function of multiple variables can be computed using the following syntax:

D[f[x,y],x]% Partial derivative with respect to x

D[f[x,y],y]% Partial derivative with respect to y

Example: To find the partial derivatives of the function , use the following commands:

D[f[x,y],x]

D[f[x,y],y]

Example:

D[x^2+y^3,x] % Outputs 2x

D[x^2+y^3,y] % Outputs 3y^2

To compute the derivative at a certain value, like use "slash-dot:"

D[f[x,y],x]/.x->4/.y->-1

##### Gradient

Compute the gradient of a function.

Grad[f[x,y],{x,y}]

Example:

Grad[x^2+y^3,{x,y}] % Outputs {2x, 3y^2}

Evaluated at

Grad[x^2+y^3,{x,y}]/.x->-3/.y->7

##### Directional Derivatives

For differentiable functions, where is a unit vector. Below is an example of computing this for a pre-existing function and a vector which is not unit length:

a=10 b=5 v={3,2} u=v/Sqrt[v.v]

Grad[f[x,y],{x,y}].u/.x->a/.y->b

##### Linearization of at a point

Recall is the linearization of at It is best to set up the right-hand side in two steps:

a=4 b=-1 Df=Grad[f[x,y],{x,y}]/.x->a/.y->b

L[x_,y_]:=f[a,b]+Dot[Df,{x-a,y-b}]

##### Critical points for

We want to find when so combine earlier techniques:

Solve[Grad[f[x,y],{x,y}]=={0,0},{x,y}]

## Unit 4 and Unit 5: Multiple Integrals |
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The following sections detail how to approach multiple integrals using Mathematica, including the usage of different coordinate systems such as rectangular, polar, and spherical coordinates.

##### Integration in Rectangular Coordinates

Following Fubini's Theorem, double and triple integrals in Mathematica can be set up as iterated integrals. For example, here are the integrals and

Integrate[Integrate[x^2 + y^2, {y, 0, x}], {x, 0, 1}]

Integrate[Integrate[Integrate[x^2 + y^2 + z^2, {z, 0, 1}], {y, 0, x}], {x, 0, 1}]

##### Integration in Polar Coordinates

The idea is the same, we just convert any integrands to polar coodinates and include the factor of In the examples below, I use t for

Integrate[Integrate[r^2, {r, 0, 1}], {t, 0, 2*Pi}] Integrate[Integrate[(r*Cos[t]+r*Sin[t])*r,{r,0,2}],{t,0,2*Pi}]

##### Integration in Spherical Coordinates

Again, same idea, just with different coordinates. I recommend using R for p for and t for Examples of triple integrals in spherical coordinates in Mathematica:

Integrate[Integrate[Integrate[R^2 Sin[p], {R, 0, 1}], {p, 0, Pi}], {t, 0, 2*Pi}]

##### Visualizing Domains in Rectangular Coordinates

We can use the RegionPlot function to visualize regions in the xy-plane and RegionPlot3D for regions in Examples:

RegionPlot[x^2 + y^2 <= 1, {x, -1, 1}, {y, -1, 1}]

RegionPlot3D[x^2 + y^2 + z^2 <= 1, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]

## Unit 6. Line and Surface Integrals |
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The following items will guide you through line and surface integrals, vector fields, and related computations in Mathematica.

##### Creating Vector Fields

Define vector fields using a function definition, but ideally with "vector inputs" (use curly brackets). Examples:

F[{x_,y_}] := {x^2, 1-y}

F[{x_,y_,z_}] := {x, y+z, Exp[x+z]}

##### Visualizing Vector Fields

VectorPlot[F[{x, y}], {x, -3, 3}, {y, -3, 3}]

StreamPlot[F[{x, y}], {x, -3, 3}, {y, -3, 3}]

##### Computing the Curl and Checking for a Conservative Field

In this unit, we compute the curl of a 2D vector field to check if it is conservative:

Curl[F[{x, y}], {x, y}] // FullSimplify

(We revisit this notion in Unit 7.)

##### Computing Scalar Line Integrals

Set up and compute scalar line integrals using a parameterized curve and a scalar function. Follow the four-step process from lecture. Example:

f[{x_,y_,z_}] := 2-z

r[t_] := {Cos[t], 0, Sin[t]}

magr = Sqrt[r'[t].r'[t]] // FullSimplify

Integrate[f[r[t]]*magr, {t, 0, Pi}]

##### Computing Vector Line Integrals

Follow the process outlined in lecture. Example setup:

F[{x_,y_}] := {2-y,x^2+y^2} r[t_] := {Cos[t], Sin[t]} Integrate[F[r[t]].r'[t], {t, 0, 2*Pi}]

##### Creating Parametric Surfaces

Create parametric surfaces using function definitions. Example:

r[u_,v_] := {u*Cos[v], u*Sin[v], u^2}

##### Computing Scalar Surface Integrals

We need to parametrize the surface, compute the magnitude of the cross product of the partial derivatives, and integrating the over the surface. Step-by-step guide based on Example 1 from **Unit 6 Lecture 13: Scalar surface integrals (15:56)**:

1-2. Define the function and the parametric surface

r[u_, v_]={u*Cos[v],u*Sin[v],9-u^2}f[{x_,y_,z_}]:=x*y

3. Compute the partial derivatives and and the the magnitude of the cross product Notice this code is re-usable for any such setup (it's all done with general variables and not specific terms):

ru=D[r[u,v],u]

rv=D[r[u,v],v]

n=Cross[ru,rv]//FullSimplify

magN=Sqrt[n.n]

4. Integrate over the bounds on and

Integrate[Integrate[f[r[u,v]]*n,{u,0,Pi/2}],{v,0,2}]

##### Computing Vector Surface Integrals

We need to parametrize the surface, compute the magnitude of the cross product of the partial derivatives, and integrating the over the surface. Step-by-step guide based on Example 1 from **Unit 6 Lecture 14: Flux Integrals (22:06)**:

1-2. Define the vector field and the parametric surface

r[u_, v_]={Cos[v],Sin[v],-u/Sqrt[1-u^2]}F[{x_,y_,z_}]:={-y,-x,1}

3. Compute the partial derivatives and and the the magnitude of the cross product Notice this code is re-usable for any such setup (it's all done with general variables and not specific terms):

ru=D[r[u,v],u]

rv=D[r[u,v],v]

n=Cross[ru,rv]//FullSimplify

(you may need to assess on your own if the above vector n is pointing in the correct direction).

4. Integrate over the bounds on and

Integrate[Integrate[F[r[u,v]].n,{u,0,1}],{v,0,2*Pi}]

## Unit 7. Vector Analysis |
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