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# Math Guides

We’ve TA’ed and tutored for many university-level courses in the past, in subjects including calculus 1, 2, 3, vector calculus, statistics, linear algebra, and differential equations, just to name a few, for the past few decades. Please check out some of the subjects below for example problems.

Feel free to contact us using the Contact Us page, and ask for help in any kind of math problem you have, we can be more than happy to help. Make sure to update Firefox or Chrome to the latest version, we use MathJax to show equations in browsers.

**Arc Length**

Arc length for multivariable functions

Arc length re-parameterization for curves

**Conservative Vector Fields – Potential and Work Calculations**

Potential and work of a conservative vector field – 1

Potential and work of a conservative vector field – 2

Potential and work of a conservative vector field – 3

Work done by conservative vector field on a particle 1

Work done by conservative vector field on a particle 2

**Determinants**

How to find determinants, 2 by 2, 3 by 3 and 5 by 5 matrices examples

**Directional Derivative**

Directional derivatives (temperature application)

**Differential Equations First-Order (ODE) Examples**

How to Classify Order and Linear or Nonlinear ODEs

Using integrating factor (first-order ODEs) 1

Using integrating factor (with integration by parts) (first-order ODEs) 2

Simplifying integrating factor (trigonometry and identities) (first-order ODEs) 3

Using integrating factor (rational function, arctan function) (first-order ODEs) 4

Newton’s Law of Cooling for “Time of Death” Forensic Applications (first-order ODEs)

Snowplow ODE Infamous Problem (first-order ODEs)

**Differential Equations Second-Order (ODE) Examples**

```
Solutions of Second-Order Homogeneous ODEs (Real, Repeated, Complex Roots)
Higher-Order Homogeneous ODEs (combinations of all real, repeated, complex roots)
Solutions of Second-Order ODES with Homogeneous and Particular solutions (Trig)
Homogeneous and Particular Solutions continued, exponential term that is repeated
Homogeneous and Particular Solutions, Hyperbolic Trigonometry term repeated
Homogeneous and Particular Solutions, Polynomial Particular Solutions
Practice Finding Particular Solutions (Method of Undetermined Coefficients) for Complicated Second-order ODEs
Using Laplace Transform and Laplace Inverse to solve Second-order ODEs (Partial Fraction Decomposition)
```

**Differential Equations – Systems of ODEs Solved Simultaneously **

```
How to Make Systems of Differential Equations for Tank Mixing Problems
Transform Second-order ODE into System of Two First-order ODEs
System of 2 ODEs Example, Real Distinct Eigenvalues and Eigenvectors
System of 3 ODEs Example, Real Distinct Eigenvalues and Eigenvectors
System of 2 ODEs Example, Complex Imaginary Eigenvalues and Eigenvectors
System of 2 ODEs Example, Repeated Real Eigenvalues and Eigenvectors
System of 3 ODEs Example, Repeated Real Eigenvalues and Eigenvectors
Solving Non-Homogeneous Systems of ODEs (Particular Solutions, Method of Undetermined Coefficients)
```

**Double Iterated Integrals – Rectangle Regions**

Double iterated integrals over rectangular regions 1

Double iterated integrals over rectangular regions 2

**Double Integrals – Transformations over Geometric Shape Regions**

Double integrals using transformations (over ellipse region)

Double integrals using transformations (over parallelogram region)

Double integrals using transformations (over triangle region)

**Double Integrals – Polar Coordinates (circular regions of integration)**

`How to Plot using Polar Coordinates`

Double integrals - how to use polar coordinate transformations

Why multiply by an extra "r" when using polar coordinates?

Double integrals using polar coordinates 1

Double integrals using polar coordinates 2

Double integrals using polar coordinates 3

Double integrals using polar coordinates (cylinder and ellipsoid regions)

Double integrals using polar coordinates (sphere and cylinder region)

**Ideal Gas Law Examples**

Ideal gas law (Avogadro’s Number and molecules) 1

Ideal gas law (mole fractions) 2

Ideal gas law (combining two gases) 3

Ideal gas law (partial pressures and mole fractions) 4

Ideal gas law (combining two gases) 5

**Jacobians**

Finding Jacobian of 3 by 3 matrix example

**Lagrange Multipliers**

Lagrange multipliers for optimization 1

Lagrange multipliers for optimization 2

Lagrange multipliers for optimization 3

**Line integrals (region of integrations of lines and curves)**

Line integrals multiple line segments 2

Line integrals parametric curves 3

**Lines and Planes Equations (Linear algebra)**

Equations for lines and planes 1

Equations for lines and planes 2

Equation for a plane from 3 points, using cross-products

**Michaelis Menten Kinetics**

Michaelis Menten kinetics fast equilibrium assumption 1

Michaelis Menten kinetics quasi/pesudo steady state assumption 2

Michaelis Menten kinetics quasi/pesudo steady state assumption 3

**Multivariable Calculus – Partial Derivatives**

**Multivariable Calculus – Chain Rule and Implicit Differentiation**

Multivariable implicit differentiation example problem

**Multivariable Tangent Plane Equations**

Tangent plane equations for multivariable functions example problem

Unit tangent vector for multivariable functions example problem

## Physics

Conservation of string in system of pulleys questions

2 Masses on an inclined slope, with friction, attached by a pulley

3 Masses on an inclined slope, with friction, attached by a pulley

Vertically stacked blocks with static and kinetic friction

Stacked blocks attached to block in pulley system

Double inclined plane, two boxes with friction in pulley system

**Second Derivative Test**

Second derivative test for multivariable function optimization and extreme point classification

**Thermodynamics – First Law Reversible Gas Expansion Practice Questions**

First Law reversible expansion, finding enthalpy, work, heat, internal energy example problem

First Law reversible expansion – isobaric and isochloric example cases

First Law reversible expansion – isothermal and adiabatic example cases

**Triple Product (cross-products and dot products)**

**Van der Waals Gas Law**

Van der Waals gas law example problem

**Velocity, Speed, and Acceleration from Displacement Equations**

Velocity, Speed, and Acceleration for Particle in 3D Motion 1

Velocity, Speed, and Acceleration for Particle in 3D Motion 2

Velocity, Speed, and Acceleration for Throwing Ball in 3D Applied Example

**Volume of Parallelpiped**

Volume of parallelpiped using triple product OR cross-products