Plan of the recitation

Calculus I / Recitation

Course goals.

Each of the following items are understood and could be calculated.

Understand the principal value of inverse trigonometric functions and evaluate inverse trigonometric functions .
Compute the derivative of inverse functions (inverse trigonometric, inverse hyperbolic functions etc).
Compute the partial derivatives of functions in several variables by the chain rule.
Compute the higher order derivatives of functions.
Understand the definition of the integral.
Find primitive functions for elementary functions.
Evaluate the double integral of functions in 2 variables.
Understand the meaning of Jacobian in the coordinate change.
Evaluate the double integral with transforming to the polar coordinates.
Determine the convergency of the improper integral.
Evaluate the improper integral.
Evaluate the improper double integral.
Evaluate the length of curves, the volume surrounded by surfaces.

Plan for each recitations and contents.

Plan for each recitations of Calculus I (Recitation)
Class	Schedule	Contents
1	Sets	Subsets, equivalency of sets, subtract of sets, product of sets, intervals, the set $R^{n}$
1	Maps	Maps, surjection, injection, bijection, composition of maps, inverse map.
2	Arithmetic functions	Exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic functions, the principal values of the inverse trigonometric functions.
2	Differentials	Differentiability, differential coefficients, derivative, derivative of composite functions, derivative of inverse function, derivative in higher order, $C^{n}$ class, Leibniz rule
3	Primitive function	Primitive function, primitive functions for elementary functions
3	Methods for primitive functions	Methods to find the primitive functions in rational, trigonometric and irrational functions.
4	Improper integral	Singularities for functions, improper integral, convergency of improper integral, absolutely convergent.
4	Functions for several variables	Limit of the values of functions, continuity on functions for several variables
5	Partial differential	Partial derivative, $C^{n}$ class functions for several variables.
5	Chain rule	Partial derivatives for composite functions, chain rule.
6	Repeated integral	Double integral, repeated integral, order change of repeated integral, Fubini's theorem.
6	Variable change in double integral	Variable change in double integral, variable change to polar coordinates, Jacobian
7	Improper double integral	Improper integral for functions in several variables, convergent sequence for domain, repeated integrals for improper integrals.
	Triple integral	Integrals in a space, 3 dimensional polar coordinates and coordinate change.
	Applications in integrals	Length of curves, area of surfaces, volume of areas.

Calculus Recitation II

Course goals.

Each of the following items are understood and could be calculated.

Determine the superior and inferior of a subset of real numbers.
Understand the convergency of sequences of real numbers and describe by $ε - N$ expression.
Determine the convergency of sequences and find its limit if it converge.
Understand the limit of value of real functions and describe by $ε - δ$ expression.
Describe the limit of value of functions of several variables by $ε - δ$ expression
Understand and describe the uniform continuity.
Find the limit of functions by using the theorem of l'Hospital.
Describe the Maclaurin (or Taylor) expansion of functions with caution to the domain of convergence.
Differentiate the implicit functions.
Find the extremal value of functions and the points it attains.
Determine the convergency of series with positive terms.
Determine the unifom convergency of sequences and series of functions.
Determine the termwise integrable or differentiable of sequences and series of functions.
Find the radius of convergent of power series.
Determine the termwise integrable or differentiable of power series.

Plan for each recitations and contents.

Plan for each recitations of Calculus Recitation II
Class	Schedule	Contents
1	Proposition	Universal proposition, existential proposition, and their negation.
	Real numbers	Bounded, superior, inferior, continuity, principal of Archimedes, density of rational numbers.
	Limits	Definition of limits ( $ε - N$ ), formulae of limits, successive limit.
2	Convergency	Criterion by the continuity, fundamental sequence, Cauchy sequence, criterion by d'Alembert, criteron by Cauchy.
2	Continuous function	Definition of limits ( $ε - δ$ )，one sided limit, convergence criterion of Cauchy.
3	Uniform continuity	Theorem of maximal-minimal value, intermediate value theorem, uniform continuity.
	Theorem of l'Hospital	Theorem of Rolle, mean value theorem, theorem of l'Hospital.
	Theorem of Taylor	Theorem of Taylor for 1 variable function, Taylor expansion, remainder term, Maclaurin expansion, polynomial approximation, symbol of Randau, extremal value for 1 variable functions.
4	Integral	Definition of definite integrals, definite integrability, fundamental theorem of calculus, indefinite integral, definition of multiple integral, area of domain, multiple integrability.
4	Integral on curves	Integrals on curves and surfaces for functions and vector fields
5	Continuous function of several variables	Limit of points, limit of functions of several variables, continuity of functions of several variables.
	Total differential	Total differential, total differentiability, directed differential, gradient of functions
	Theorem of Taylor for several variables	Taylor and Maclaurin expansion for several variables, implicit function theorem.
6	Extremal value	Extremal value, extremal value with conditions, Hessian.
6	Positive term series	Criteria of convergency for positive term series, comparison test, d'Alembert's test, Cauchy's test, theorem of Leibnits, absolutely convergent, conditional convergent.
7	Function sequence	Sequence of functions, limit function, uniform convergent, series of functions, convergency criteria, M-criteria of Weierstrass, exchange limits and differential or integral, termwise integral, termwise differential.
7	Power series	Domain of convergent of power series, radius of convergent, theorem of Abel, termwise integral and termwise differential of power series and its radius of convergent.

Linear Algebra I / Recitation

Course goals.

Each of the following items are understood and could be calculated.

Calculate sum and product of matrices, and understand the definition of regular matrices.
Solve systems of linear equations by applying the row reduction
Find the inverse matrix by applying the row reduction.
Understand the rank of matrices, and determine the existence of the solution of linear equations, their free variables, regularity of matrices by rank.
Calculate the determinant of matrices.
Determine the regularity of matrices by its determinant and express the inverse matrix by the adjoint matrix.
Describe the solution of linear equations by the theorem of Cramer.

Plan for each recitations and contents.

Plan for each recitations of Linear Algebra I (Recitation)
Class	Schedule	Contents
1	Vectors and matrices	Matrix as a representation of linear transformation of vectors.
2	Matrix	Definition of matrix, operation of matrices, zero matrix, unit matrix, transposed matrix, diagonal matrix.
2	Regular matrix	Regular matrix, definition of the inverse matrix.
3	Partition of matrix	Partition to smaller matrices, symmetric partition, vector partition, elementary vector.
3	Elementary transformation	Elementary transformation of matrices, elementary matrices.
4	Echelon matrix	Elementary transformation of partitioned matrices, echelon matrices, row reduction.
4	System of linear equations	Solution of linear equations, condition for existence of the solution, free variable of the solution, solution space.
5	Inverse matrix	Calculus of inverse matrix.
5	Rank	Rank of matrices, canonical rank form, relation with rank and regularity.
6	Determinant	Definition of determinant, determinant and elementary transformation, relation with determinant and regularity.
7	Adjoint expansion	Adjoint of matrices, adjoint matrix, adjoint expansion of determinant, formula of Cramer.

Linear Algebra Recitation II

Course goals.

Each of the following items are understood and could be calculated.

Understand the general vector spaces.
Find basis and dimension of vector space.
Understand the linear map and describe its representation matrix.
Determine the matrix of the base change.
Determine the representation matrix when base changed.
Find the image and the kernel of linear maps.
Calculate complex inner product and find orthonormal basis.
Find the eigenvalue, eigenvector, eigenspace of matrices.
Determine the diagonalizability of matrices, and diagonalize if available.
Diagonalize the normal matrices by unitary matrices.
Transform real normal matrices to canonical form by orthogonal matrices.

Plan for each recitations and contents.

Plan for each recitations of Linear Algebra Recitation II
Class	Schedule	Contents
1	Vector space	Definition of vector space, linear combination of vectors, linear independent, linear dependent.
2	Vector subspace	Sum and intersection of vector subspaces, vector subspace generated by vectors.
2	Basis and dimension	Basis of vector space, dimension, standard basis, dimension theorem of subspaces.
3	Linear map	Definition of linear map, isomorphism, isomorphic vector spaces.
3	Basis and coordinate	Coordinate respected to basis, transformation matrix for base change.
4	Representation matrix	Representation matrix of linear map respected to basis, representation matrix related with base change.
4	Image and kernel	Image and kernel of linear maps, dimension theorem for linear map.
5	Inner product	Definition of inner product, norm of vector, orthogonality, orthogonal complement.
5	Orthogonalization	Schumidt's orthogonalization, orthonormal basis, Hermite matrix, unitary matrix.
6	Eigenvalue and eigenvector	Eigenvalue, eigenvector, eigenspace, characteristic polynomial, multiplicity of eigenvalue.
6	Diagonalization	Formula of Ceyley-Hamilton, diagonalizability and diagonalization of matrices
7	Normal matrix	Diagonalize normal matrices by unitary matrices.
7	Canonical form of real normal matrix	Canonical form of real normal matrices, canonical form by orthogonal matrices.

Back to the Guide to the Mathematics Courses.