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# Advanced Calculus / Multivariable Calculus

### Syllabus

1_1 Exponential Growth and Decay.flv.
1_2 Exponential Growth and Decay.flv.
1_3 Exponential Growth and Decay.flv.
1_4 Exponential Growth and Decay.
1_5 Euler Method.
1_6 Euler Method.
2_1 Sequences.
2_2 Sequences.
2_3 Sequences.
2_4 Sequences.
3_1 Introduction to Series.
3_1_1 Introduction to Series.
3_2 The Geometric Series.
3_3 The Harmonic Series.
3_4 Example Problems Involving Series.
3_5_1 The Integral Test and Comparison Tests.
3_5_2 The Integral Test and Comparison Tests.
3_5_3 The Integral Test and Comaprison Tests.
3_5_4 The Integral Test and Comparison Tests.
3_6_2 Alternating Series.
3_6_3 Alternating Series.
3_7_1 Absolute Value Test.
3_7_2 Ratio and Root Tests.flv.
4_1_1 Power Series.
4_1_2 Power Series.
10_1_1 Vector Function Differentiation.
10_1_2 Examples of Vector Function Differentiation.
10_1_3 Examples of Vector Function Differentiation.
11_1_1 Introduction to the Differentiation of Multivariable Functions.
11_1_2 Example Problems on Partial Derivative of a Multivariable Function.
11_2_1 The Geomtery of a Multivariable Function.
11_3_1 The Gradient of a Multivariable Function.
11_3_2 Working towards an equation for a tangent plane to a multivariable point.
11_3_4 Working towards an equation for a tangent plane to a multivariable function.
11_3_5 When is a multivariable function continuous.
11_3_6 Continuity and Differentiablility.
11_3_7 A Smooth Function.
11_3_8 Example problem calculating a tangent hyperplane.
11_4_1 The Derivative of the Composition of Functions.
11_4_2 The Derivative of the Composition of Functions.
11_5_1 Directional Derivative of a Multivariable Function Part 1.
11_5_2 Directional Derivative of a Multivariable Function Part 2.
11_6_1 Contours and Tangents to Contours Part 1.
11_6_2 Contours and Tangents to Contours Part 2.
11_6_3 Contours and Tangents to Contrours Part 3.
11_7_1 Potential Function of a Vector Field Part 1.
11_7_2 Potential Function of a Vector Field Part 2.
11_7_3 Potential Function of a Vector Field Part 3.
11_8_1 Higher Order Partial Derivatives Part 1.
11_9_1 Derivative of Vector Field Functions.
11_9_2 Conservative Vector Fields.
12_1_1 Introduction to Taylor Polynomials.
12_1_2 An Introduction to Taylor Polynomials.
12_1_3 Example problem creating a Taylor Polynomial.
12_2_1 Taylor Polynomials of Multivariable Functions.
12_2_2 Taylor Theorem for Multivariable Polynomials.
13_1 An Introduction to Optimization in Multivariable Functions.
13_2 Optimization with Constraints.
14_1 The Double Integral.
14_2 The Type I Region.
14_3 Type II Region with Solved Example Problem.
14_4 Some Fun with the Volume of a Cylinder.
14_5 The double integral calculated with polar coordinates.
14_6 Changing between Type I and II Regions.
14_7 Translation of Axes.
14_8 The Volume of a Cylinder Revisited.
14_9 The Volume between Two Functions.
14_10 The Triple Integral by way of an Example Problem.
14_11 The Translation of Axes in Triple Integrals.
14_12 Translation to Cylindrical Coordinates.
15_1 An Introduction to Line Integrals.
15_2_1 Example Problem Explaining the Line Integral with Respect to Arc Length.
15_2_2 Another Example Problem Solving a Line Integral.
15_2_3 Another example problem without using a parametrized curve.
15_3_1 Line integrals with respect to coordinate variables.
15_3_2 Example problem with line integrals with respect to coordinate variables.
15_3_3 Continuation of previous problem.
15_4_1 Example problem with the line integral of a multivariable functions.
15_4_2 Example problem with the line integral of a multivariable functions.
15_4_3 Example problem with the line integrals of a multivariable functions.
16_1 Introduction to line integrals of vector fields.
16_2 Evaluating the force and the directional vector differential.
16_3 Example problem solving the line integral of a vector field.
16_4 Another example problem solving for the line integral of a vector field.
16_5 Another example problem solving for the line integral of a vector field.
16_6 Another problem solving for the line integral in a vector field.
16_7 The fundamental theorem of line integrals.
16_8 The line integral over a closed path.
17_1 The surface integral.
17_2 Example problem solving for the surface integral.
18_1 Introduction to flux.
18_2 Calculating the normal vector.
18_3 Example problem for flux.
19_1 Greens Theorem.
19_1_2 Example problem using theorem of Green to solve for a line integral.
19_1_3 Another example problem solving for the line integral using the theorem of Green.
19_2 The Theorem of Stokes.
19_2_1 Example problem using the theorem of Stokes.
19_3_1 Example problem using the theorem of Gauss.
19_3_2 Example problem using theorem of Gauss.
Understanding the Euler Lagrange Equation.

Dr Juan Klopper

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