LaTex files for assignments from UVic MATH 202, covering topics such as:
- Vector Geometry: Vector operations, normalization, dot and cross products, and geometric applications.
- 3D Geometry: Lines, planes, distances, intersections, and skew lines in three dimensions.
- Multivariable Functions: Domains, limits, continuity, and partial derivatives in two or more variables.
- Differential Equations: First-order ODEs including separable, linear, exact, and Bernoulli equations.
- Partial Differential Equations: Verification of solutions to PDEs such as the heat equation using multivariable chain rule.
- Existence and Uniqueness: Conditions for existence and uniqueness of solutions to initial value problems.
- Laplace Transforms: Forward and inverse Laplace transforms, shift properties, and solving IVPs.
- Higher-Order ODEs: Second-order linear ODEs with constant coefficients, variation of parameters, and Cauchy-Euler equations.
- Proof Techniques: Rigorous proofs involving vector identities, gradient vectors, and ODE solution properties.
- Wind velocity and aircraft motion problems.
- Vector magnitude, normalization, and resultant velocity calculation.
- Boat towing and direction problems.
- Dot product and angle calculations between force vectors.
- Cross product and 3D geometry.
- Parallelogram area and parallelepiped volume using cross and scalar triple products.
- Vector identity proof.
- Triple scalar product and trigonometric identity verification.
- Distance from a point to a line in 3D.
- Cross product formula for point-to-line distance.
- Volume of a parallelepiped.
- Triple scalar product via determinant computation.
- Line-plane intersection.
- Parametric line equations and substitution into plane equations.
- Perpendicular planes.
- Normal vectors and dot product perpendicularity condition.
- Domain of a multivariable function involving arcsin.
- Constraint inequalities and annular domain in polar coordinates.
- Distance between skew lines.
- Cross product of direction vectors and shortest distance projection formula.
- Multivariable limits.
- Polar coordinate substitution and path-dependent limit analysis to show non-existence.
- Removable discontinuity of a multivariable function.
- Factoring and defining a piecewise continuous extension.
- Gradient and directional derivatives.
- Computing partial derivatives and the direction of steepest ascent.
- Verification of the heat equation.
- Computing mixed partial derivatives using the multivariable chain rule.
- Existence and uniqueness theorem for ODEs.
- Checking continuity of f and ∂f/∂y to identify regions of guaranteed unique solutions.
- Separable first-order ODE.
- Separation of variables and integration by parts.
- First-order linear ODEs and exact equations.
- Integrating factor method, initial value problems, and potential function construction.
- Piecewise ODE with continuity matching at boundary.
- Bernoulli equation.
- Substitution to reduce to linear form and back-substitution of solution.
- Exact differential equation.
- Exactness verification and potential function construction via partial integration.
- Bernoulli equation with fractional exponent.
- Substitution v = y^(2/3), reduction to linear form, and re-expression of solution.
- Laplace transforms.
- Shift property and multiplication-by-t property.
- Inverse Laplace transforms.
- Completing the square and inverse transforms of exponential-cosine forms.
- Solving second-order ODEs with Laplace transforms.
- Applying initial conditions, partial fraction decomposition, and inverse transforms.
- Variation of parameters.
- Wronskian computation and particular solution via parameter variation.
- Cauchy-Euler equation.
- Characteristic equation via y = x^m substitution and handling repeated roots.