FOUNDATION COURSES ENGINEERING MATHEMATICS
II
UNIT I LAPLACE
TRANSFORM
Laplace
transform – Sufficient Condition for existence – Transform of elementary
functions – Basic properties – Transform of derivatives and integrals ––
Transform of periodic functions -
Inverse Laplace transform– Convolution theorem (excluding proof) –
Initial and Final value theorems - Solution of linear ODE of second order with
constant coefficients using Laplace transform.
UNIT II VECTOR DIFFERENTIAL CALCULUS
Vector
fields and scalar fields - The gradient field - The directional derivative-
Divergence and Curl of a vector field- Solenoidal and Irrotational vector
fields- The Laplacian in polar, cylindrical, and spherical coordinates.
UNIT III
VECTOR INTEGRAL CALCULUS
Line
integrals in the plane-Line integrals as integrals of vectors- Green’s theorem
(without proof) in the plane and its verification- Line integrals in space-
Surfaces in space- Normal to the surface- Orientability- Surface integrals-
Divergence theorem (without proof) and Stokes’ theorem (without proof) and
their verification involving cubes and rectangular parallelepiped only.
UNIT IV
ANALYTIC FUNCTIONS
Functions of a complex variable – Analytic functions – Necessary conditions,
Cauchy – Riemann equation and Sufficient conditions (excluding proofs) –
Harmonic and orthogonal properties of analytic function – Harmonic conjugate –
Construction of analytic functions – Conformal mapping : w= z+c, cz, 1/z, and
bilinear transformation.
UNIT V
COMPLEX INTEGRATION
Complex
integration – Statement and applications of Cauchy’s integral theorem and
Cauchy’s integral formula – Taylor and Laurent expansions – Singular points –
Residues – Residue theorem – Application of residue theorem to evaluate real
integrals –Unit circle and semi-circular
contour(excluding poles on boundaries).
Online resources
https://www.khanacademy.org
http://ceee.rice.edu
UNIT I LAPLACE TRANSFORM
Laplace
transform – Sufficient Condition for existence – Transform of elementary
functions – Basic properties – Transform of derivatives and integrals ––
Transform of periodic functions -
Inverse Laplace transform– Convolution theorem (excluding proof) –
Initial and Final value theorems - Solution of linear ODE of second order with
constant coefficients using Laplace transform.
UNIT II VECTOR DIFFERENTIAL CALCULUS
Vector
fields and scalar fields - The gradient field - The directional derivative-
Divergence and Curl of a vector field- Solenoidal and Irrotational vector
fields- The Laplacian in polar, cylindrical, and spherical coordinates.
UNIT III
VECTOR INTEGRAL CALCULUS
Line
integrals in the plane-Line integrals as integrals of vectors- Green’s theorem
(without proof) in the plane and its verification- Line integrals in space-
Surfaces in space- Normal to the surface- Orientability- Surface integrals-
Divergence theorem (without proof) and Stokes’ theorem (without proof) and
their verification involving cubes and rectangular parallelepiped only.
UNIT IV
ANALYTIC FUNCTIONS
Functions of a complex variable – Analytic functions – Necessary conditions,
Cauchy – Riemann equation and Sufficient conditions (excluding proofs) –
Harmonic and orthogonal properties of analytic function – Harmonic conjugate –
Construction of analytic functions – Conformal mapping : w= z+c, cz, 1/z, and
bilinear transformation.
UNIT V
COMPLEX INTEGRATION
Complex
integration – Statement and applications of Cauchy’s integral theorem and
Cauchy’s integral formula – Taylor and Laurent expansions – Singular points –
Residues – Residue theorem – Application of residue theorem to evaluate real
integrals –Unit circle and semi-circular
contour(excluding poles on boundaries).
Online resources
https://www.khanacademy.org
http://ceee.rice.edu
0 Comments
Post a Comment