**Multivariable
Calculus/Differential Equations**

**Multivariable Calculus**

Example of Equation of a Sphere

Equation of a Sphere Given Diameter

Equation of Sphere Given Tangent Plane 1

Equation of Sphere Given Tangent Plane 2

Equation of Sphere Given Tangent Plane 3

Angle Between Two Vectors Using Dot Product

Vector Decomposition of (2,2,1) Along (1,1,1)

Unit Vector Perpendicular to Two Vectors

Area of Parallelogram in Three Space

Diagonal Lengths of a Parallelepiped

Example of Symmetric Equations of a Line

Equation of a Plane Containing a Point and a Line

Equation of a Plane Through Three Points

Planes: Parallel, Equal, or Intersecting?

Line of Intersection of Two Planes

Example of Plane Line Intersections

Domain of a Vector-Valued Function

Limit and Derivative of Vector Function

Example of Position, Velocity and Acceleration in Three Space

Tangent Line to a Parametrized Curve

Angle of Intersection Between Two Curves

Unit Tangent and Normal Vectors for a Helix

Sketch/Area of Polar Curve r = sin(3O)

Arc Length along Polar Curve r = e^{-O}

Showing a Limit Does Not Exist

Contour Map of f(x,y) = 1/(x^2 + y^2)

Sketch of a One-Sheeted Hyperboloid

Sketch of a Double-Napped Cone

Gradient of f(x,y) = yx^2 + cos(xy)

Example of Implicit Differentiation with Several Variables

Tangent Plane to x^2 - xy - y^2 -z = 0

Lagrange Multiplier: Single Constraint

Optimization on Ellipse in R^3 1: Parametrization Method

Optimization on Ellipse in R^3 2: Lagrange Multipliers with Two Constraints

Example of Chain Rule for Partial Derivatives

Second Partials Test for f(x,y) = x^3 + 3xy + y^3

Directional Derivative of f(x,y,z) = xz + yz

Linear Approximation to f(x,y) = x^2y^2 + x

Taylor Polynomial of f(x,y) = ycos(x+y)

Conversion From Rectangular Coordinates

Conversion From Cylindrical Coordinates

Conversion from Spherical Coordinates

Examples of Double and Triple Integrals

Center of Mass for a Rectangle of Variable Density

Interchange of Limits of Integration

Area Between Polar Curves r = 2/cos(O) and r = 4cos(O)

Integral of exp(-x^2) (HD Version)

Surface area of z = (x^2+y2)^1/2

Mass of Solid as a Triple integral in Rectangular Coordinates

Volume of Truncated Paraboloid in Cylindrical Coordinates

Volume of a Snow Cone in Cylindrical and Spherical Coordinates

Example of Arc Length Along a Parametrized Curve

**Differential Equations**

General Solution of y' + xy = 0

Verifying the Solution of an ODE

The Logistic Function 1: Solving The ODE

The Logistic Function 2: Sketching The S-Curve

General Solution to y' - 3y = b(x)

Example of Population Growth 1

Example of Population Growth 2

Example of Radioactive Decay 1

Example of Radioactive Decay 2

General Solution to y'' - 6y' + 9y = 0

Complex Numbers for ODEs (1 of 4)

Complex Numbers for ODEs (2 of 4)

Complex Numbers for ODEs (3 of 4)

Complex Numbers for ODEs (4 of 4)

General Solution to y'' - 6y' +25y = 0

Antiderivative of e^{3x} cos(4x) (ODE Solution)

Antiderivative of x^2 e^x (ODE Solution)

General Solution of y'''-4y''+5y'-2y=0

Wronskian for {e^{3x}, e^{-x}, 2}

Linear Dependence of {x^2-1, x^2+x, x+1}

Annihilator Method 1: Real Linear Factors

Example of Annihilator Method: y''-y = sin(2x)

Power Series Solution for y''-2y'+y=x, y(0)=0, y'(0)=1

Example: Solving ODE by Substitution

Mass-Spring Systems 1: Free, Undamped Motion

Mass-Spring Systems 2: Free, Underdamped Motion

Mass-Spring Systems 3: Critically Damped Motion

Laplace Transform of f(t) = 2t-1

Laplace Transform of f(t) = sin(2t)

Laplace Transform of f(t) = t sin(2t)

Laplace Transform of f(t) = e^{3t}cos(4t)

Laplace Transform of f(t) = t^2 e^{2t} cos(3t)

Inverse Laplace Transform of (s-1)/s^2(s^2+4)

Laplace Transform Solution of y'-3y=e^{2t}, y(0)=2

Laplace Transform Solution of y''-2y'-3y=e^t, y(0) = 0, y'(0) = 1

Laplace Transform of f(t) = 2 on the Interval (1,2)

Second Shift Formula for a Piecewise-defined Function

Laplace Transform Solution of y'-y=f(t) (Piecewise-Defined)

Example of Convolution Theorem: f(t)=t, g(t)=sin(t)

Convolution Theorem for y'-2y=e^t, y(0)=0