Section 13.2: Calculus of vector-valued functions
Paul's notes
Videos
Derivative of vector-valued functions
Properties
More properties
Equation of tangent line
Angle of two curves
Integration with Initial Conditions
Definite integral
Section 13.1: Vector-valued functions
Paul's notes
Here are 3 different web-based 3D graphing programs:
Desmos 3D,
GeoGebra 3D calculator, and
CalcPlot3D app.
Feel free to let me know about other apps.
Here is a helix made with Desmos 3D.
Here is are two surfaces in \(\mathbb{R}^3\).
Find a parameterization of the intersection.
Mathematica is available to you with a site license.
Here is a Mathematica notebook, helix.nb,
that will plot a parameterized curve in \(\mathbb{R}^3\).
It is harder to use than some other programs, but the pictures are beautiful.
This Mathematica notebook will Plot 2 Surfaces And 1 Curve.
Videos
Vector valued functions
Domain of a Vector Valued Function
space curves in 3D Calc Plotter
Curve of intersection of two surfaces
Curve of intersection of two surfaces
Vector Valued Function from a Rectangular Equation
Section 12.7: Cylindrical and Spherical Coordinates.
Here are Paul's notes on
polar,
cylindrical, and
spherical coordinates.
BEWARE! Paul lies to you. He writes the formula “\(\theta = \tan^{-1} (\frac y x )\)”,
which is wrong in two ways: (1) That formula is fine in quadrant I but it is false in quadrants II and III,
and (2) it uses the abominable notation “\(\tan^{-1}\)” instead of “\(\arctan\)”.
\(f(\theta) = \tan(\theta)\) is not a one-to-one function so its inverse is not defined. Instead, \(\arctan(t)\) is the unique \(\theta\)
between \(-\pi/2\) and \(\pi/2\) whose tangent is \(t\).
Instead, write is
“\(\tan(\theta) = \frac y x\)”
To compute \(\theta\) you need to draw a darn picture!
Here is a figure about how to compute theta in polar or cylindrical
or spherical coordinates using the picture you have drawn.
For those who prefer a formula,
I wrote this algorithm for finding \(\theta\).
Note that \(\frac y x\) is undefined if \(x = 0\), so those cases are handled first.
I don't intend you humans to follow this algorithm exactly: Just draw the #!#! picture.
BEWARE! The roles of \(\theta\) and \(\phi\) are switched in physics and math. We are using the math convention, which is superior. In our math notation, the coordinate \(\theta\) is the same in polar, cylindrical and spherical coordinates: \(\tan(\theta) = \frac y x\).
Videos
Cylindrical Coordinates
Spherical Coordinates
Cartesian Coordinates to Spherical
Spherical Coordinates to Cartesian
Cartesian Coordinates to Cylindrical
Cylindrical Coordinates to Cartesian
Cylindrical Equations to Rectangular
Rectangular Equations to Cylindrical
Spherical Equations to Rectangular
Rectangular Equation to a Spherical
Spherical Equation to a Rectangular
Section 12.6: Quadratic Surfaces
Paul's notes
Here are some quadratic surfaces
plotted in desmos 3d.
Another option is using the "implicit plot" feature of
CalcPlot3D
CalcPlot3D has more features than desmos 3d, but is consequently harder to use.
Videos
how to plot surfaces in 3D Calc Plotter.
The following videos have more detail than you need. They use the term "quadric surface" which is another name for "quadratic surface". You do not need to learn the names of these surfaces.
Cylindrical Surfaces
Quadric Surfaces
Ellipsoid
Elliptical Cone
Elliptical Paraboloid
Hyperbolic Paraboloid
Section 12.5: Planes in space
Paul's notes
Desmos graph for the line in \(\mathbb{R}^2\) through the point (2, 3) with the normal vector (2, -1).
An equation is
2(x-2) - (y-3) = 0
Desmos graph for the plane \(\mathbb{R}^3\) through point (-1, 2, 1) with the normal vector (2, -1, 3). An equation is 2(x+1) - (y-2) + 3(z-1) = 0
Videos
Normal equation of plane
Point of Intersection of a Plane and a Line
Point of Intersection of a Plane and a Line
Intersection to two planes
Line through a point and perpendicular to a plane
Plane given with point and parallel plane
Plane given with three points
Plane given with three points
Plane given with point and orthogonal line
Angle between two planes
Distance between point and plane
Distance between parallel planes
Distance between line and point
Section 12.4: The Cross Product.
Paul's notes
Example: calculation of the cross product done in two ways.
This is my problem 6 on the WeBWorK. You only need to do it one way, but it's comforting that they
give the same answer.
Suggested videos on matrices
Multiplying matrices
2x2 determinant
3x3 determinant
Suggested videos on the cross product
Cross product
Cross product
Cross product example
Area of space triangle
Volume of parallelepiped
Section 12.3: The Dot Product.
Paul's notes
Suggested videos.
Dot product
Angle between vectors
Parallel and perpendicular components
In-class group work. The methane molecule \(C H_4\) has 4 Hydrogen atoms at the vertices
of a tetrahedron, with a single carbon atom at the center.
We can place the carbon atom at \( (0,0,0) \), and the 4 Hydrogen atoms at
\( (1,1,1)\), \((1, -1,-1)\), \( (-1,1, -1)\), and \(( -1,-1,1) \).
This desmos3D graph shows a
Methane molecule.
,
Find the bond angle, which
is the angle between any two vectors from the carbon atom to a hydrogen atom.
First get an exact angle in terms of the arccos function, then approximate the angle
to the nearest degree using technology.
Solutions.
Here are some formulas needed for the webwork. Only some are in Paul's Notes.
It is easiest to make sense of these formulas using the notation \(\hat{\bf a}\) (pronounced a-hat) for the unit vector in the direction of \(\bf a\). To compute that unit vector, use the formula \(\hat{\bf a} = \frac{\bf a}{\|\bf{a}\|}\).
The component of \(\bf b\) along \(\bf a\) is \(\hat{\bf a} \cdot {\bf b} = \frac{\bf a \cdot {\bf b}}{\| {\bf a} \|} = \| \bf b \| \cos(\theta)\), which is a scalar.
The projection of \(\bf b\) parallel to \(\bf a\) is \({\bf b}_{\| {\bf a}} = (\hat{\bf a} \cdot {\bf b}) \hat {\bf a} = \frac{({\bf a} \cdot {\bf b}) {\bf a}}{\| {\bf a} \|^2}\), which is a vector. Note that the final expression has no square roots.
The projection of \(\bf b\) perpendicular to \(\bf a\) is \({\bf b}_{\perp {\bf a}} = {\bf b} - {\bf b}_{\| {\bf a}} \), which is a vector.
Note that \({\bf b} = {\bf b}_{\| {\bf a}} + {\bf b}_{\perp {\bf a}} \) must hold (by definition), and it turns out that
\({\bf b}_{\| {\bf a}} \cdot {\bf b}_{\perp {\bf a}} = 0\), so these two projections are orthogonal to each other.
Here is a scan of a version of set 12.3, problem 9, with nice numbers. This example should hopefully give some intuition about the parallel and orthogonal projections of vectors.
Section 12.2: Vectors in \(\mathbb{R}^3\)
Paul's notes
Suggested videos.
Parametric equation of line in 3D
Parametric equation of line in 3D
Intersection of two lines
Section 12.1: Vectors (Note: The section number 12.1 follows the Rogowski and Adams textbook.)
Paul's notes on
basics and
vector arithmetic.
Here is my version of webwork problem 5.
The answer to one version of WeBWorK problem 10 in set 12.1 is
x = a + 1.6 b. (So type 1 and 1.6 into the two blanks.)
This figure shows the vectors.
Here is a Mathemetica notebook to allow arbitrary linear combinations of
a and b.
Suggested videos.
Vector operations: Sum, scalar multiple, dot product
Length of a 3D vector
Unit vectors: Direction of a vector
Plotting points
WeBWorK set 01: Review of Calculus
Differentiation Shortcuts
Here is the Big Picture of Calculus.
