Multivariable Calculus
Multivariable Calculus
Table of Contents
- Vectors & Analytic Geometry in Space
- Intro to 3D Equations
- HMT & Magnitude
- Vector Operations
- Resolving a 2D Vector
- Dot Product
- Vector Projection
- Work
- Cross Product
- Parallelogram (Cross Product)
- Vector Function for a Line
- Plane Perpendicular to a Vector
- Distance (Point to Line)
- Distance (Point to Plane)
- Distance (Two Skew Lines)
- Cylinders and Generating Surfaces
- Quadric Surfaces
- Vector Valued Functions & Motion in Space
- Unit Tangent Vector
- Multivariable Functions & their Derivatives
- Contour Maps
- Partial Derivatives
- Partial Derivatives Vector Visualization
- Clairaut's Theorem Demo
- Tangent Planes
- Directional Derivative
- Gradient and Velocity
- Min, Max, and Saddle Points
- Concavity
- 2nd Partial Derivatives Test Visualization
- Lagrange Multipliers
- Multiple Integrals
- Double Integrals
- Double Integral Calculator (Change Order of Integration)
- Polar Review
- Triple Integrals
- Triple Integrals (Changing Integration Order)
- Equations in Cylindrical Coordinates
- Triple Integrals w/Cylindrical Coordinates
- Equations in Spherical Coordinates
- Spherical Graphs Illustrator
- Triple Integral w/Spherical Coordinates
- Jacobian Demo
- Integration in Vector Fields
- Smooth Curve Parametrization Practice
- Line Integrals
- Vector Fields
- Gradient Vector Fields
- Flux and Flow
- Conservative Fields
- Curl & Divergence Visualization
- Surface Integrals
- Surface Flux
Intro to 3D Equations
Instructions
Every question below has an applet below for you to use as you experiment with the 3-Dimensioanl space. [br]Once you have answered the question, you may click on "Check My Answers" to see the correct responses.
What is the equation of the [i]xy[/i]-plane in the Three Dimensional Space? What about the [i]zy-[/i]plane?
What does the graph of x=2 look like when viewed on a number line (Singular Dimension)?
What does the graph of x=2 look like when viewed in the 2 Dimensional Cartesian plane?
What does the graph of x=2 look like when viewed in 3 dimensions?
[list=1][*]What’s the equation of a plane [i][b]parallel [/b][/i]to the[i][b] xy-plane[/b][/i] and contains the point (3, 4, 8)?[/*][*]What’s the equation of a plane [i][b]parallel [/b][/i]to the [b][i]xz-plane[/i][/b] and contains the point (1, 6, 5)?[/*][*]What's the equation of a plane [i][b]perpendicular [/b][/i]to the [i][b]yz-plane[/b][/i] and contains the point (1, 6, 5)[/*][*]What's the equation of a plane [i][b]perpendicular [/b][/i]to the [i][b]xz-plane[/b][/i] and contains the point (1, 6, 5)?[/*][/list][br][br]
What does the graph of [math]x^2+y^2=16[/math] look like when viewed in the Cartesian plane?
Describe the graph of [math]f\left(x\right)=sin\left(x\right)[/math] in the Cartesian Plane.
The last two graphs you described are created in the Cartesian plane, in other words, they are 2-dimensional. What do the same graphs look like when examined in 3 dimensions? [br]Make your conjectures and test them using the applet below.
Describe the intersection of [math]x^2+y^2=9[/math] and [math]z=5[/math].
Describe the graph of each of the following[br][list=1][*][math]x^2+y^2+z^2=9,z\le0[/math][br][/*][*][math]x^2+y^2+z^2\le9[/math][br][/*][*][math]x^2+y^2+z^2<9[/math][br][br][br][/*][/list]
Find the equation(s) for a circle with radius 3 centered at [math]\left(-2,5,1\right)[/math] and lying in a plane parallel to the [i]yz[/i]-plane.
Find the equation(s) of a with radius 2 centered at [math]\left(-2,6,-3\right)[/math] and lying in a plane perpendicular to the [i]xz-[/i]plane.