This course covers first-order separable differential equations, sequences, series, and multivariable calculus. Students will learn about exponential growth and decay, Euler method, power series, vector function differentiation, partial derivatives, optimization, double and triple integrals, line integrals, surface integrals, flux, Green's theorem, Stokes' theorem, Gauss' theorem, and the Euler Lagrange Equation. The course uses a combination of video lectures and example problems to teach these advanced calculus concepts. The intended audience for this course is learners who have a strong foundation in calculus and are looking to deepen their understanding of multivariable calculus and its applications.
Overview
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.
Taught by
Dr Juan Klopper
