# MAT2214

The textbook for this course is Calculus (Volume 3) by Gilbert Strang, Edwin Herman, et al.

## Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
```1.1
```
• Plot points using polar coordinates and find several polar coordinates of a single point
• Convert polar coordinates to rectangular coordinates and vice versa
• Transform equations from polar form to rectangular form and vice versa
Week 1
```1.2
```

• Three-dimensional coordinate systems.
• Distance Formula in Space.
• Standard Equation for a Sphere.
Weeks 1/2
```2.1
```
• Vector Algebra Operations
• The Magnitude of a vector
• Unit Vectors
• The Midpoint of a Line Segment
• The Vector projection
Week 2
```2.3
```

• Definition of Dot Product
• Properties of Dot Product
• Angle between vectors
• Orthogonal vectors
Week 2
```2.4
```
• Definition of Cross Product
• Properties of the cross product
• Area of a parallelogram
• Cross product as a determinant

Week 3
```2.5
```
• Write the vector, parametric equation of a line through a given point in a given direction, and a line through two given points.
• Find the distance from a point to a given line.
• Write the equation of a plane through a given point with a given normal, and a plane through three given points.
• Find the distance from a point to a given plane.

Week 3
```2.6
```
• Find equations for cylinders that are generated by rotating lines that are parallel to a plane

Weeks 3/4
```3.1, 3.2
```
• Vector functions
• Limits of vector functions
• Continuity of vector functions
• Differentiation rules for vector functions
• Curves and paths in space

Week 4
```3.3
```
• The arc Length of a vector function
• Arc length parameterization
Weeks 4/5
```3.4
```
• The Unit tangent vector
• The curvature
• The Principal Unit Normal Vector
• The Binormal Vector
• The tangential and normal components of acceleration
• The Torsion
Week 5/6
```4.1
```
• Functions of two variables
• Functions of three variables
• Domain and range of multivariable functions
• Bounded regions
• Graphs and level curves of two variable functions
• Level surfaces of three variable functions
Week 6
```4.2
```
• Limits of functions of two variables
• Limits of functions of more than two variables
• Properties of limits of functions of several variables
• Two path test of non-existing of a limit
• Continuity for functions of several variables
• Continuity of composition
• Extreme values on closed and bounded domains
Week 6
```4.3
```

• Partial derivatives for functions of two variables
• Partial derivatives for functions of more than two variables
• Partial derivatives and continuity
• Second order partial derivatives
• Mixed derivative theorem
Week 7
```4.4
```
• Directional derivatives for functions of two variables
• Properties of directional derivatives
• Tangents to level curves
• Directional derivatives for functions of three variables
Week 7
```4.5
```

• Determine the equation of a plane tangent to a given surface at a point
• Determine the parametric equation of a normal line to a given surface at a point
• The linear approximation of a function of two variables at a point
• The definition of differentiability for a function of two variables
• Differentiability implies Continuity
• Continuity of First Partial Derivatives implies Differentiability
• The definition of total differentiability for a function of two variables
• Use the total differential to approximate the change in a function of two variables
Week 7
```4.6
```
• Chain rule for functions of one independent variable and several intermediate variables.
• Chain rule for functions of two independent variable and several intermediate variables.
• Method for implicit differentiation.
• The general chain rule for functions of several independent variables
Week 8
```4.7
```
• The derivative test for local extreme values
• Extreme values on closed and bounded domains
• Critical points and saddle points for functions of two variables
• Second derivative test for local extreme values
• Absolute maxima and minima on closed and bounded regions
Week 8/9
```4.8
```
• Lagrange Multipliers with One Constraint
• Lagrange Multipliers with Two Constraints
Week 9/10
```5.1
```
• Double Integral is the limit of Double Sums.
• Double Integrals over Rectangular Regions.
• Interated Integrals.
• Fubini's Theorem (part 1).
Week 10
```5.2
```
• Double integrals over bounded, general regions.
• Properties of double Integrals.
• Fubini's theorem (part 2)
• Changing the order of Integration.
• Calculating Volumes, Areas and Average Values
Week 11
```5.3
```
• Double Integrals over rectangular polar regions.
• Double Integrals over general polar regions.
• Changing Cartesian Integrals into Polar Integrals.
• Using Double Integrals in Polar Coordinates to find Volumes, Areas.
Week 11
``` 5.4
```
• Triple Integrals over general bounded regions.
• Finding Volumes by evaluating Triple Integrals.
• Average value of a function in space.
• Changing Integration Order and Coordinate systems.
Week 12
```5.5
```
• Integrations in Cylindrical Coordinates.
• Equations relating rectangular and cylindrical coordinates.
• Changing Cartesian integrations into Cylindrical integrations.
• Integrations in Spherical coordinates.
• Equations relating spherical coordinates to Cartesian and cylindrical coordinates.
• Changing Cartesian integrations into Cylindrical integrations.
Week 13
```5.6
```
• Finding Masses, Moments, Centers of Masses, Moments of Inertia in Two Dimensions.
• Finding Masses, Moments, Centers of Masses, Moments of Inertia in Three Dimensions.
Week 13/14
```5.7
```
• Determine the image of a region under a given transformation of variables.
• Compute the Jacobian of a given transformation.
• Evaluate a double integral using a change of variables.
• Evaluate a triple integral using a change of variables.
Week 14
```6.1
```
• Vector Fields in a plane.
• Vector Fields in Space.
• Potential Functions.
• Gradient Fields, Conservative Vector Fields.
• The Cross-Partial Test for Conservative Vector Fields.
• Determining Whether a Vector Field is conservative.
Week 14
```6.2
```
• Line Integrals of functions a long a smooth curves in a planer or in space
• Line Integrals of of vector fields along an oriented curves in a plane or space..
• Properties of Vector Line Integrals.
• Evaluating Line Integrals.
• Applications of line integrals: Calculating Arc Length, the Mass of a wire, Work done by a force, Flux across a curve, Circulation of a force.
Week 14/15
```6.3
```
• Describe simple and closed curves
• Define connected and simply connected regions.
• Explain how to test a vector field to determine whether it is conservative.
• Find a potential function for a conservative vector field.
• Use the Fundamental Theorem for Line Integrals to evaluate a line integral of a vector field.
Weeks 14/15
```6.4
```
• Circulation form of Green's Theorem.
• Flux Form of Green’s Theorem.
• Applying Green's Theorem to find Work, Flux.