Notes
- Prof: Dr Tatiana Pilehrood
- TA: Tamis Panagiotakis
parker.glynn.adey@utoronto.cat.hessamipilehrood@utoronto.ca- Tutorial Schedule: https://q.utoronto.ca/courses/405948/pages/tutorial-schedule
Concepts
Week 1
- Graph
- Limits Across Paths
- Multi-Variate Function
- Graph Slice
- Level Sets
- Partial Derivative
- Euclidean Space
Week 2
- Vector
- Parametric Equation
- Parametric Equation from Two Points
- Symmetric Equation
- Symmetric Equation from Two Points
- Inner Product and Angles
- Normal Vector
- Cartesian Equation of Plane
- Finding Equation of Plane by Point and Normal Vector
- Angle Between Two Planes
- Intersection Between Two Planes
- Normal Vector from Cartesian Equation of Plane
- Projection
- Distance Between Point to Line
- Distance Between Point and Plane
- Point on Plane Closest to Point
- Cross Product
- Triple Product Identity
- Normal Vector from Two Vectors of Plane
Week 3
- Graph
- Contour Map
- Sphere
- Ellipsoid
- Paraboloid
- Saddle
- Hyperboloid
- Cone
- Open Ball
- Open Set
- Ball-Box Inequality
- Limits
- Limits Open Set Definition
- Epsilon Delta Limit Definition
- Two Dimensional Limit Proof Example
- Limits Across Paths
- Limits Existence Implies Equality Along All Paths
- Continuity
- Partial Derivative
- Linear Approximation
- Component Functions
- Derivative Matrix Definition
- Gradient
- Differentiability
- Linear Approximation and Differentiability Theorem
Week 4
- Product Rule for Derivatives
- Chain Rule for Multivariable Function
- Chain Rule for Total Derivatives
- Chain Rule and Gradients
- Directional Derivative
- Directional Derivative and Gradients
- Gradient as the Direction of Fastest Increase
- Gradient Perpendicular to Level Sets
- Hessian Matrix
Week 5
- Unit Normals to a Curve
- Necessary Conditions for Differentiability
- Pure Partial Derivative
- Iterated Partial Derivative
- Class Functions
- Clairauts Theorem
- Subscript Notation for Derivatives
- Heat Equation
- Wave Equation
- Differentiability
- First-Order Multivariable Taylor Series
- Second-Order Multivariable Taylor Series
- Multi-Index
- General Multivariable Taylor Series
- Taylor’s Theorem in One Dimension
- Taylor’s Theorem in One Dimension Constant Version
- Taylor Polynomial to Find Partial Derivative
- Quadratic Form
- Quadratic Form Representation of Matrix
- Hessian Matrix
Week 6
- Local Extrema
- Global Extrema
- Critical Points
- Saddle Point
- First Derivative Test for Local Extrema
- Hessian Function
- Positive Definite Matrix
- Test for Definiteness
- Second Derivative Test for Local Extrema
- Minimize Distance Squared
- Disk
- Bounded
- Boundary Point
- Closed Set
- Extreme Value Theorem
- Minima and Maxima for Sets with Boundary Process
- Plane Tangent to Surface at Point
- Tangent Plane to Line
Week 7
- Restriction Operator
- Objective Function
- Lagrange Multiplier Method
- Lagrange Auxilliary Function
- Level Curves Interpretation of Lagrange
- Lagrange With Multiple Constraints
- Arithmetic-Geometric Mean Inequality
Week 8
- Cavalieri’s Principle
- Double Integral Volume
- Definite Integral Riemann Sum Definition
- Riemann Integral in Two Dimensions
- Bounded Function
- Proving Integral is Bounded
- Iterated Integrals
- Fubini’s Theorem
- Definite Integral
Week 9
- Y-Simple Region
- X-Simple Region
- Simple Region
- Double Integral for Y-Simple Region
- Change of Order of Integration
- Mean Value Theorem for Integrals
- Multiple Integral Splitting Theorem