Presentation on theme: "VECTOR CALCULUS 1.10 GRADIENT OF A SCALAR 1.11 DIVERGENCE OF A VECTOR"— Presentation transcript:
1VECTOR CALCULUS 1.10 GRADIENT OF A SCALAR 1.11 DIVERGENCE OF A VECTOR DIVERGENCE THEOREMCURL OF A VECTORSTOKES’S THEOREMLAPLACIAN OF A SCALAR
21.10 GRADIENT OF A SCALAR Suppose is the temperature at , and is the temperature atas shown.
3GRADIENT OF A SCALAR (Cont’d) The differential distances are the components of the differential distance vector :However, from differential calculus, the differential temperature:
4GRADIENT OF A SCALAR (Cont’d) But,So, previous equation can be rewritten as:
5GRADIENT OF A SCALAR (Cont’d) The vector inside square brackets defines the change of temperature corresponding to a vector change in positionThis vector is called Gradient of Scalar T.For Cartesian coordinate:
6GRADIENT OF A SCALAR (Cont’d) For Circular cylindrical coordinate:For Spherical coordinate:
7EXAMPLE 10Find gradient of these scalars:(a)(b)(c)
8SOLUTION TO EXAMPLE 10(a) Use gradient for Cartesian coordinate:
9SOLUTION TO EXAMPLE 10 (Cont’d) (b) Use gradient for Circular cylindricalcoordinate:
10SOLUTION TO EXAMPLE 10 (Cont’d) (c) Use gradient for Spherical coordinate:
11DIVERGENCE OF A VECTORIllustration of the divergence of a vector field at point P:Positive DivergenceNegative DivergenceZero Divergence
12DIVERGENCE OF A VECTOR (Cont’d) The divergence of A at a given point P is the outward flux per unit volume:
13DIVERGENCE OF A VECTOR (Cont’d) Vector field A at closed surface SWhat is ??
14DIVERGENCE OF A VECTOR (Cont’d) Where,And, v is volume enclosed by surface S
15DIVERGENCE OF A VECTOR (Cont’d) For Cartesian coordinate:For Circular cylindrical coordinate:
16DIVERGENCE OF A VECTOR (Cont’d) For Spherical coordinate:
17EXAMPLE 11Find divergence of these vectors:(a)(b)(c)
18SOLUTION TO EXAMPLE 11(a) Use divergence for Cartesian coordinate:
19SOLUTION TO EXAMPLE 11 (Cont’d) (b) Use divergence for Circular cylindricalcoordinate:
20SOLUTION TO EXAMPLE 11 (Cont’d) (c) Use divergence for Spherical coordinate:
21DIVERGENCE THEOREMIt states that the total outward flux of a vector field A at the closed surface S is the same as volume integral of divergence of A.
22EXAMPLE 12A vector field exists in the region between two concentric cylindrical surfaces defined by ρ = 1 and ρ = 2, with both cylinders extending between z = 0 and z = 5. Verify the divergence theorem by evaluating:(a)(b)
23SOLUTION TO EXAMPLE 12 (a) For two concentric cylinder, the left side: Where,
26SOLUTION TO EXAMPLE 12 Cont’d) (b) For the right side of Divergence Theorem,evaluate divergence of DSo,
27CURL OF A VECTORThe curl of vector A is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.
29CURL OF A VECTOR (Cont’d) The curl of the vector field is concerned with rotation of the vector field. Rotation can be used to measure the uniformity of the field, the more non uniform the field, the larger value of curl.
30CURL OF A VECTOR (Cont’d) For Cartesian coordinate:
31CURL OF A VECTOR (Cont’d) For Circular cylindrical coordinate:
32CURL OF A VECTOR (Cont’d) For Spherical coordinate:
38STOKE’S THEOREMThe circulation of a vector field A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L that A and curl of A are continuous on S.