Syllabus
1. Introduction into vector calculus. Scalar and vector. Position vector. Linear combination, linear dependence of vectors. Representation of vectors in terms of base vectors. Coordinate systems in plane and space. Matrices and determinants.
2. Scalar product - definition, geometric interpretation and properties. Magnitude of a vector, angle between vectors, vector coordinates. Applications in physics - work, area vector. Vector product - definition, geometric interpretation and properties. Axial vector, rotations, angular velocity vector.
3. Scalar triple product and vector triple product - definition, geometric interpretation and properties. Right-hand and left-hand coordinate system of general vectors. Reciprocal vector. Summary of vector algebra.
4. Vector transformations, dyad - definition and properties. Physical situations leading to tensors. Definition of tensor, tensor algebra, symmetric and antisymmetric tensors. Quadric and covariant of tensor.
5. Introduction into vector analysis. Vector function of scalar argument - definition, limit, derivation and primitive function.
6. Scalar and vector functions of vector argument - scalar and vector field. Partial and total derivative. Vector form of total derivative - Hamilton nabla operator.
7. Properties of Hamilton nabla operator. Vector operations with Hamilton operator - divergence, rotation and gradient of vector. Exercises. Operations of second order. Exercises.
8. Flow of a vector through a surface. Gauss theorem.
9. Rotation of a vector along a curve. Stokes theorem. Potential and non-potential field.
10. Application of vector analysis in physical problems. Maxwell equations in differential form. Wave equation for planar electromagnetic wave, relation between vectors E, B and k.
Annotation
Repetition of basic concepts and operations of vector calculus. Applications of vector algebra in physical situations.
Introduction of tensors, elementary properties and operations with tensors. Introduction into vector analysis, scalar and vector functions of vector argument.
Hamilton nabla operator. Concept of gradient, divergence and rotation of vector function, physical interpretation.
Gauss and Stokes theorem of vector analysis, applications in Maxwell equations.