MATH-106(e): Analysis II

Lectures in this course (103)

Explores directional derivatives, tangent planes, and normal vectors in surfaces.

Explains partial and directional derivatives, and functions' derivability.

Covers the calculation of partial derivatives of order 3 and the concept of differentiability in functions.

Explores functions of class C^p, Hessian matrices, and methods to verify differentiability at a point.

Explores the properties and methods of differentiable functions, including strong recurrence and demonstration techniques.

Explores functions with values in R^m, gradients, derivatives, and the Jacobian matrix in multiple dimensions.

Explains the Jacobian matrix and derivative of composite functions with examples.

Explores the derivative of composite functions and their application in changing variables.

Explores the derivative of an integral with parameter dependency and its continuity.

Covers methods of proof, such as recurrence on two variables and the gradient.

Explores Taylor's formula in two variables, emphasizing its application and importance for functions of a single variable.

Covers the Taylor polynomial in two variables and the concept of extrema in functions of several variables.

Explores the conditions for local extrema of functions in several variables, including critical points and the Hessian matrix.

Explains local extremum conditions for n=2 and n=3, critical points, and stationary points.

Discusses finding min and max of functions on compact sets and the theorem of implicit functions.

Explains the Implicit Functions Theorem for n=2 and n=3 cases.

Explores finding the equation of a tangent hyperplane to a function and its implications.

Explores the Lagrange Multipliers Theorem, covering extrema conditions and geometric interpretations.

Covers the application of Lagrange multipliers to find extrema of functions under constraints.

Explains firm pavements, Darboux sums, and the conditions for a function to be integrable.

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