Calculus Revisited: Calculus of Complex Variables, Differential Equations, and Linear Algebra
by Herbert Gross
To listen to an audio podcast, mouse over the title and click Play. Open iTunes to download and subscribe to iTunes U collections.
This course gives an introduction to Complex Variables, Ordinary Differential Equations and Linear Algebra.
|1||VideoPart I: Complex Variables, Lecture 1: The Complex Numbers||Herb Gross explains the need to define complex numbers. He defines the structure of the system of complex numbers including addition, subtraction, multiplication, division, powers and roots and shows that the system is closed under all these operations.||1/10/2012||Free||View In iTunes|
|2||VideoPart I: Complex Variables, Lecture 2: Functions of a Complex Variable||Professor Herb Gross discusses functions of a complex variable, limits, derivatives and the Cauchy-Riemann conditions. Functions of a complex variable that are differentiable everywhere are called analytic functions.||1/10/2012||Free||View In iTunes|
|3||VideoPart I: Complex Variables, Lecture 3: Conformal Mappings||Herb Gross defines and explains what is meant by a conformal mapping.||1/10/2012||Free||View In iTunes|
|4||VideoPart I: Complex Variables, Lecture 4: Sequences and Series||Herb Gross defines complex valued functions by means of power series expansions. He shows us how the amazing identity of DeMoivre is derived.||1/10/2012||Free||View In iTunes|
|5||VideoPart I: Complex Variables, Lecture 5: Integrating Complex Functions||Herb Gross generalizes the definition of the integral of a real-valued function of a real variable to the integral of a complex-valued function of a complex variable and examines the ramifications of this generalization.||1/10/2012||Free||View In iTunes|
|6||VideoPart II: Differential Equations, Lecture 1: The Concept of a General Solution||Herb Gross defines and illustrates the different types of solutions of a differential equation: General solutions, particular solutions and singular solutions.||1/10/2012||Free||View In iTunes|
|7||VideoPart II: Differential Equations, Lecture 2: Linear Differential Equations||Herb Gross defines and illustrates linear differential equations of order 2. Herb also shows how to find solutions of this type of equation.||1/10/2012||Free||View In iTunes|
|8||VideoPart II: Differential Equations, Lecture 3: Solving the Linear Equations L(y) = 0; Constant Coefficients||Herb Gross talks about a specific type of Differential Equations, namely those that are linear, 2nd order, homogeneous and with constant coefficients. He gives examples of three types of possible general solutions and then shows why they ARE the solutions||1/10/2012||Free||View In iTunes|
|9||VideoPart II: Differential Equations, Lecture 4: Undetermined Coefficients||Herb Gross shows how to find particular (and general) solutions of second order linear differential equations with constant coefficients using the method of undetermined coefficients.||1/10/2012||Free||View In iTunes|
|10||VideoPart II: Differential Equations, Lecture 5: Variations of Parameters||Herb Gross uses the method of Variation of Parameters to find a particular solution of linear homogeneous order 2 differential equations when the general solution is known.||1/10/2012||Free||View In iTunes|
|11||VideoPart II: Differential Equations, Lecture 6: Power Series Solutions||Herb Gross show how to find the general solution of a liner, homogeneous differential equation of order 2 when the coefficients are analytic functions.||1/10/2012||Free||View In iTunes|
|12||VideoPart II: Differential Equations, Lecture 7: Laplace Transforms||Herb Gross describes and justifies the use of LaPlace Transforms as a method of solving linear differential equations with given initial conditions.||1/10/2012||Free||View In iTunes|
|13||VideoPart III: Linear Algebra, Lecture 1: Vector Spaces||Herb Gross describes and illustrates the axiomatic definition of a vector space and discusses subspaces.||1/10/2012||Free||View In iTunes|
|14||VideoPart III: Linear Algebra, Lecture 2: Spanning Vectors||Herb Gross describes spanning vectors: Vectors whose linear combinations generate a vector space. Herb also defines linear dependence and the dimension of a vector space.||1/10/2012||Free||View In iTunes|
|15||VideoPart III: Linear Algebra, Lecture 3: Constructing Bases||Herb Gross gives a short review of bases and dimension. He then does an example using the row-reduced matrix technique.||1/10/2012||Free||View In iTunes|
|16||VideoPart III: Linear Algebra, Lecture 4: Linear Transformations||Herb Gross defines Linear Transformations from vector space V into vector space W. He also defines and gives examples of the null space of such a map and illustrates the matrix representation of a linear transformation relative to a given basis.||1/10/2012||Free||View In iTunes|
|17||VideoPart III: Linear Algebra, Lecture 5: Determinants||Herb Gross examines the determinant in the framework of vector spaces. He shows us how the value of a particular determinant tells us whether a given set of vectors is a basis of a given vector space with a known basis.||1/10/2012||Free||View In iTunes|
|18||VideoPart III: Linear Algebra, Lecture 6: Eigenvectors||Herb Gross defines an eigenvector of a linear map f as a vector x that is mapped into a constant multiple, c, of itself. The value of c is called the eigenvalue (or characteristic) for the corresponding vector x.||1/10/2012||Free||View In iTunes|
|19||VideoPart III: Linear Algebra, Lecture 7: Dot Products||Herb Gross axiomatically defines the dot product as the map of ordered pairs of vectors into the real numbers. Using this definition, Herb next defines and shows how and why to find an orthonormal basis.||1/10/2012||Free||View In iTunes|
|20||VideoPart III: Linear Algebra, Lecture 8: Orthogonal Functions||Herb Gross defines and illustrates the Fourier representation of a piecewise continuous function.||1/10/2012||Free||View In iTunes|
There has to be said about how MIT has Professors that have a true passion on teaching... Their lectures expand on so much more... A joy to watch, just for fun...
Viewers also subscribed to
- Category: Calculus
- Language: English
- http://ocw.mit.edu; Creative Commons Attribution-NonCommercial-ShareAlike 3.0; http://ocw.mit.edu/OcwWeb/web/terms/terms; Album art photo by Christopher Ariza.