1. Tensor calculus. Scalar, vector, tensor. Algebra of vectors, differential operations on vectors.
2. Cartesian tensors of second-rank. Algebra of tensors, differential operations on tensors. Tensor
diagonalization.
3. Field theory. The gradient of scalar fields. Scalar potential. The divergence and rotation in vector fields.
The curl theorem due to Stokes. The divergence theorem due to Gauss.
4. Orthogonal curvilinear coordinates. The gradient, divergence and curl in cylindrical and spherical
systems. An application of transport phenomena: Equation of continuity. Heat and mass transfer.
5. Boundary value problems for ordinary differential equation (ODE). Formulation of boundary value
problems. Boundary conditions. Second-order linear differential equations. Orthogonal system of
functions. Fourier series. Homogeneous boundary problem. The eigenvalues and eigenfunctions.
6. Sturm-Liouville problem. The Bessel’s differential equation. Some method of solution of non-
homogeneous boundary value problems. Method of direct integrations. Fourier method.
7. Method of variation of constants. Finite difference method. Application: Stationary heat conduction
equation. Stationary, one-dimensional heat transfer in a plane sheet, cylinder and sphere.
8. Boundary value problems for partial differential equation (PDE). Linear second-order homogeneous
partial differential equations and their classification. Boundary conditions, initial conditions. Formulation
of boundary value problems for parabolic and elliptic equations. Some method of solutions of boundary
value problem.
9. The method of separation of variables. Application: The one-dimensional heat conduction equation.
Non-dimensional variables.
10. The method of combination of variables. Application: The one-dimensional diffusion equation in semi-
infinite medium.
11. The method of Green’s function. Application: The one-dimensional heat conduction equation in infinite
medium.
12. The difference method. Finite-difference solution: explicit method, Crank-Nicolson implicit method.
Compatibility, convergence, stability.
13. Solution of the two-dimensional Laplace equation: The method of separation of variables. Finite
difference method.
14. Reserve
2. Cartesian tensors of second-rank. Algebra of tensors, differential operations on tensors. Tensor
diagonalization.
3. Field theory. The gradient of scalar fields. Scalar potential. The divergence and rotation in vector fields.
The curl theorem due to Stokes. The divergence theorem due to Gauss.
4. Orthogonal curvilinear coordinates. The gradient, divergence and curl in cylindrical and spherical
systems. An application of transport phenomena: Equation of continuity. Heat and mass transfer.
5. Boundary value problems for ordinary differential equation (ODE). Formulation of boundary value
problems. Boundary conditions. Second-order linear differential equations. Orthogonal system of
functions. Fourier series. Homogeneous boundary problem. The eigenvalues and eigenfunctions.
6. Sturm-Liouville problem. The Bessel’s differential equation. Some method of solution of non-
homogeneous boundary value problems. Method of direct integrations. Fourier method.
7. Method of variation of constants. Finite difference method. Application: Stationary heat conduction
equation. Stationary, one-dimensional heat transfer in a plane sheet, cylinder and sphere.
8. Boundary value problems for partial differential equation (PDE). Linear second-order homogeneous
partial differential equations and their classification. Boundary conditions, initial conditions. Formulation
of boundary value problems for parabolic and elliptic equations. Some method of solutions of boundary
value problem.
9. The method of separation of variables. Application: The one-dimensional heat conduction equation.
Non-dimensional variables.
10. The method of combination of variables. Application: The one-dimensional diffusion equation in semi-
infinite medium.
11. The method of Green’s function. Application: The one-dimensional heat conduction equation in infinite
medium.
12. The difference method. Finite-difference solution: explicit method, Crank-Nicolson implicit method.
Compatibility, convergence, stability.
13. Solution of the two-dimensional Laplace equation: The method of separation of variables. Finite
difference method.
14. Reserve