EED 2008 : Electromagnetic Theory

1. EED 2008: Electromagnetic Theory VectorsDivergence and Stokes Theorem Özgür TAMER

2. Vector integration • Linear integrals • Vector area and surface integrals • Volume integrals

3. Line Integral • The line integral is the integral of the tangential component of A along Curve L • Closed contour integral (abca) Circulation of A around L A is a vector field

4. Surface Integral (flux) • Vector field A containing the smooth surface S • Also called; Flux of A through S • Closed Surface Integral Net outward flux of A from S A is a vector field

5. Volume Integral • Integral of scalar over the volume V

6. Vector Differential Operator • The vector differential operator (gradient operator), is not a vectorin itself, but when it operates on a scalar function, for example, a vector ensues.

7. Gradient Physical meaning of T : A variable position vector r to describe an isothermal surface : Since dr lies on the isothermal plane… and Thus, T must be perpendicular to dr. Since dr lies in any direction on the plane, T must be perpendicular to the tangent plane at r. • if A·B = 0 • The vector A is zero • The vector B is zero •  = 90° T T is a vector in the direction of the most rapid change of T,and its magnitude is equal to this rate of change. dr

8. Gradient 1- Definition. f(x,y,z) is a differentiable scalar field 2 – Physical meaning: is a vector that represents both the magnitude and the direction of the maximum space rate of increase of Φ

9. Divergence The operator  is of vector form, a scalar product can be obtained as : A is the net flux of A per unit volume at the point considered, counting vectors into the volume as negative, and vectors out of the volume as positive. Output - input : the net rate of mass flow from unit volume

10. Divergence Ain Aout The flux leaving the one end must exceed the flux entering at the other end. The tubular element is “divergent” in the direction of flow. Therefore, the operator  is frequently called the “divergence” : Divergence of a vector

11. Divergence 1 – Definition is a differentiable vector field x x+dx 2 – Physical meaning The divergence of A at a given point P is the outward flux per unit volume as thevolume shrinks about P.

12. Divergence (a) Positivedivergence, (b) negative divergence, (c) zero divergence.

13. Divergence • To evaluate the divergence of a vector field A at point P(x0,y0,x0), we let the point surrounded by a differential volume After some series expansions we get;

14. Divergence • Cylindrical Coordinate System • Spherical Coordinate System

15. Divergence • Properties of the divergence of a vector field • It produces a scalar field • The divergence of a scalar V, div V, makes no sense

16. 1 – Definition.The curl of a is an axial (or rotational) vector whose magnitude is the maximum circulationof A per unit area as the area lends to zero and whose direction is the normaldirection of the area when the area is oriented so as to make the circulationmaximum. • Curl

17. Curl • 2 – Physical meaning: is related tothe local rotation of the vectorfield: If is the fluid velocity vectorfield

18. B  y v u A O  x Curl What is its physical meaning? Assume a two-dimensional fluid element Regarded as the angular velocity of OA, direction : k Thus, the angular velocity of OA is ; similarily, the angular velocity of OB is

19. B  y v u A O  x Curl The angular velocity of the fluid element is the average of the two angular velocities :  This value is called the “vorticity” of the fluid element, which is twice the angular velocity of the fluid element. This is the reason why it is called the “curl” operator.

20. Curl • Cartesian Coordinates

21. Curl • Cylindrical Coordinates

22. Curl • Spherical Coordinates

23. Stokes’ Theorem Considering a surface S having element dS and curve C denotes the curve : If there is a vector field A, then the line integral of A taken round C is equal to the surface integral of  × A taken over S : Two-dimensional system

24. Stokes’ Theorem • Stokes's theorem states that ihe circulation of a vector field A around a (closed) pth L is equal to the surface integral of the curl of A over the open surface S bounded byL provided that A and are continuous on S

25. Laplacian 1 – Scalar Laplacian.The Laplacian of a scalar field V, written as. is the divergence of the gradientof V. The Laplacian of a scalar field is scalar Gradient of a scalar is vector Divergence of a vector is scalar

26. Laplacian • In cartesian coordinates • In Cylindrical coordinates • In Spherical Coordinates

27. As a second derivative, the one-dimensional Laplacianoperator is related to minima and maxima: when the second derivative is positive (negative), the curvature is concave (convexe). f(x) • Laplacian: physical meaning convex concave x In most of situations, the 2-dimensional Laplacianoperator is also related to local minima and maxima. If vE is positive: E

28. Laplacian • A scalar field V is said to be harmonic in a given region if its Laplacian vanishes inthat region.

29. Laplacian • Laplacian of a vector: is defined as the gradient of the divergence of A minus the curl of the curl ofA; • Only for the cartesian coordinate system;

30. 3. Differential operators • Summary resp.

31. Gauss’ Divergence Theorem • The divergence theorem states thatthe total outward flux of a vector field A through the closed surface S is the same as the volume integral of the divergence of A • The theorem applies to any volume v bounded by theclosed surface S

32. Gauss’ Divergence Theorem Ain Aout The tubular element is “divergent” in the direction of flow. The net rate of mass flow from unit volume We also have : The surface integral of the velocity vector u gives the net volumetric flow across the surface The mass flow rate of a closed surface (volume)

33. Stokes’ Theorem Gauss’ Divergence Theorem

34. Classification of Vector Fields • A vector field is characterized by its divergence and curl

35. Classification of Vector Fields • Solenoidal Vector Field: A vector field A is said to be solenoidal (or divergenceless) if • Such a field has neither source nor sink of flux, flux lines of A entering any closed surface must also leave it.

36. Classification of Vector Fields • A vector field A is said to be irrotational (or potential) if • In an irrotational field A, the circulation of A around a closed path is identically zero. • This implies that the line integral of A is independent of the chosen path • An irrotationalfield is also known as a conservative field

37. Stokes formula: vector field global circulation Theorem. If S(C)is any oriented surfacedelimited by C: S(C) C Sketch of proof. y Vy . Vx . . x P . … and then extend to any surface delimited by C.

38. Divergence Formula: global conservation laws Theorem. If V(C)is the volumedelimited by S Sketch of proof. Flow through the oriented elementary planes x = ctt and x+dx = ctt: x x+dx -Vx(x,y,z).dydz + Vx(x+dx,y,z).dydz and then extend this expression to the lateral surface of the cube. Other expression: extended to the vol. of the elementary cube: