1. Vector integration Linear integrals Vector area and surface integrals
Volume integrals

2. 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

3. 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

4. Volume Integral
Integral of scalar over the volume V

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

6. Gradient Physical meaning of  T :
A variable position vector r to describe an isothermal surface : dr T
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.
T is a vector in the direction of the most rapid change of T, and its magnitude is equal to this rate of change.
if A·B = 0
The vector A is zero
The vector B is zero
 = 90°

7. 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

8. 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

9. 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

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

11. Divergence
(a) Positive divergence, (b) negative divergence, (c) zero divergence.

12. 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;

13. Divergence
Cylindrical Coordinate System
Spherical Coordinate System

14. 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

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

16. Curl
2 – Physical meaning: is related to the local rotation of the vectorfield: If
is the fluid velocity vectorfield

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

18. Curl
The angular velocity of the fluid element is the average of the two angular velocities : u v
 x
 y O A B 
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.

19. Curl
Cartesian Coordinates

20. Curl
Cylindrical Coordinates

21. Curl
Spherical Coordinates

22. 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

23. 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 by L provided that A and are continuous on S

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

25. Laplacian In cartesian coordinates In Cylindrical coordinates
In Spherical Coordinates

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

27. Laplacian
A scalar field V is said to be harmonic in a given region if its Laplacian vanishes in that region.

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

29. 3. Differential operators
Summary
Operator
grad
div
curl
Laplacian is
a vector
a scalar
(resp. a vector)
concerns
a scalar field
a vector field
(resp. a vector field)
Definition
resp.

30. Gauss’ Divergence Theorem
The divergence theorem states that the 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 the closed surface S

31. 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)

32. Stokes’ Theorem
Gauss’ Divergence Theorem

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

34. 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.

35. 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 irrotational field is also known as a conservative field

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

37. Divergence Formula: global conservation laws
Theorem. If V(C) is the volume delimited 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: