"Advanced Concepts in Vector Calculus: Exploring Gradients, Divergence, and Curl Through Multivariable Functions and Differential Geometry"

Vector calculs basic for better understanding for engineering

1. Vector Calculus

2. Electromagnetic problems involve integration along a curve, over a surface or throughout a volume. Line Integrals : Line integral of a vector A along a path situated in a curve in space is given by It is the integral of the tangential component of A along L. If the path of integration is a closed path is called the circulation of A around L. Line, Surface and Volume Integrals

3. Surface Integrals: Surface integral of A through a surface S is given by Where  represents the flux of A through S. For a closed surface is the net outward flow of flux from S Line, Surface and Volume Integrals ( Contd ) Volume Integral The volume integral of the scalar 𝞺 v over the volume v is defined as Q= To evaluate the integrations the differential elements in length, area and volume must be known.

4. Differential Elements-Cartesian Coordinates

5. 1- 5 Differential Elements in Cylindrical Coordinates

6. Differential elements-cylindrical

7. 1- 7 Differential Elements in Spherical Coordinates

8. Del operator Known as gradient operator Cartesian Cylindrical Spherical

9. Gradient of a scalar Cartesian Cylindrical Spherical

10. Gradient of scalar- Numericals

11. Divergence of a vector Positive negative zero The divergence is defined as a measure how much field diverges or emanates or originates from a particular point

12. Understanding divergence Vector is a function which has 2 dimensional inputs and gives 2 D outputs Divergence of vector is a new function which takes 2D point as input and output depends on behaviour of field in a small neighbourhood around that point

13. Expression for divergence of D In order to evaluate the integral over the closed surface, the integral must be broken up into six integrals, one over each face.

14. Expression for Divergence of D Combining the results of all the integrals we get, From this we can write As a limit , The last term is 𝞺 v , hence

15. Contn … We define the del operator as In terms of the del operator, we can define the divergence as

16. Divergence in other Coordinates In Cylindrical coordinates In spherical co-ordinates Properties of divergence field

17. Divergence Theorem Divergence of field equation Let v be the volume divided into large number of small cells. Let nth cell have volume The outward flux through surface Sn is: The total outward flux will be the addition of flux through all cells.

18. Thus the divergence theorem relates surface integral to volume integral The integral of the normal component of any vector field over a closed surface is equal to the integral of the divergence of this vector field throughout the volume enclosed by the closed surface.  

19. Numericals

20. Contd..

21. Numericals

22. Numericals

23. Contd..

24. Numericals

25. CURL OF VECTOR The curl of A is an axial (or rotational) vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum

26. CURL OF VECTOR

27. CURL OF VECTOR in Cartesian

28. Curl of vector in cylindrical

29. Curl of vector in spherical

30. Properties of curl

31. STOKES THEOREM From curl, we understood, R elates a vector surface integral to a line integral around the boundary of a surface The Stoke's theorem states that the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface. Divergence theorem relates surface integral to Volume integral. Stoke’s theorem related line integral (Circulation) to surface integral

32. Numericals on curl and divergence

33. Laplasian of scalar