Course on Vector Calculus

Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.

The objective of the course is to introduce and develop the methods of vector analysis. These methods provide a natural aid to the understanding of geometry and some physical concepts. They are also a fundamental tool in many theories of Applied Mathematics.

After completion of the course, students will have adequate background, conceptual clarity and knowledge of mathematical principles related to Vector differentiation

By the end of the course, students should be able to:

• Calculate scalar and vector products.

• Find the vector equations of lines and planes.

• Understand the parametric equations of curves and surfaces.

• Differentiate vector functions of a single variable.

• Calculate velocity and acceleration vectors for moving particles.

• Understand and be able to find the unit tangent vector, the unit principal normal and the curvature of a space curve.

• Find the gradient of a function.

• Find the divergence and curl of a vector field and prove identities involving these.

• Use the gradient operator to calculate the directional derivative of a function.

• Calculate the unit normal at a point on a surface.

• Recognise irrotational and solenoidal vector fields.

• Evaluate line and surface integrals.

• Understand the various integral theorems relating line, surface and volume integrals.

Few applications of Vector Differential Calculus

Because vectors and matrices are used in linear algebra, anything that requires the use of arrays that are linear dependent requires vectors. A few well-known examples are:

1. Internet search
2. Graph analysis
3. Machine learning
4. Graphics
5. Bioinformatics
6. Scientific computing
7. Data mining
8. Computer vision
9. Speech recognition
10. Compilers
11. Parallel computing