![]() SOLUTION TO EXAMPLE 13 (Cont’d) (c) Use curl for Spherical coordinate:ġ.14 STOKE’S THEOREM The 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.ĮXAMPLE 14 By using Stoke’s Theorem, evaluate for SOLUTION TO EXAMPLE 13 (Cont’d) (b) Use curl for Circular cylindrical coordinate SOLUTION TO EXAMPLE 13 (a) Use curl for Cartesian coordinate: Rotation can be used to measure the uniformity of the field, the more non uniform the field, the larger value of curl.ĬURL OF A VECTOR (Cont’d) For Cartesian coordinate:ĬURL OF A VECTOR (Cont’d) For Circular cylindrical coordinate:ĬURL OF A VECTOR (Cont’d) For Spherical coordinate:ĮXAMPLE 13 Find curl of these vectors: (a) (b) (c) SOLUTION TO EXAMPLE 12 Cont’d) (b) For the right side of Divergence Theorem, evaluate divergence of D So,ġ.13 CURL OF A VECTOR The 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.ĬURL OF A VECTOR (Cont’d) The curl of the vector field is concerned with rotation of the vector field. SOLUTION TO EXAMPLE 12 (a) For two concentric cylinder, the left side: Where, ![]() Verify the divergence theorem by evaluating: (a) (b) SOLUTION TO EXAMPLE 11 (Cont’d) (c) Use divergence for Spherical coordinate:ġ.12 DIVERGENCE THEOREM It 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.ĮXAMPLE 12 A 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. SOLUTION TO EXAMPLE 11 (Cont’d) (b) Use divergence for Circular cylindrical coordinate: SOLUTION TO EXAMPLE 11 (a) Use divergence for Cartesian coordinate: SOLUTION TO EXAMPLE 10 (Cont’d) (c) Use gradient for Spherical coordinate:ġ.11 DIVERGENCE OF A VECTOR Illustration of the divergence of a vector field at point P: Positive Divergence Negative Divergence Zero DivergenceĭIVERGENCE OF A VECTOR (Cont’d) The divergence of A at a given point P is the outward flux per unit volume:ĭIVERGENCE OF A VECTOR (Cont’d) Vector field A at closed surface S What is ?ĭIVERGENCE OF A VECTOR (Cont’d) Where, And, v is volume enclosed by surface SĭIVERGENCE OF A VECTOR (Cont’d) For Cartesian coordinate: For Circular cylindrical coordinate:ĭIVERGENCE OF A VECTOR (Cont’d) For Spherical coordinate:ĮXAMPLE 11 Find divergence of these vectors: (a) (b) (c) SOLUTION TO EXAMPLE 10 (Cont’d) (b) Use gradient for Circular cylindrical coordinate: SOLUTION TO EXAMPLE 10 (a) Use gradient for Cartesian coordinate: GRADIENT OF A SCALAR (Cont’d) For Circular cylindrical coordinate: For Spherical coordinate:ĮXAMPLE 10 Find gradient of these scalars: (a) (b) (c) ![]() This vector is called Gradient of Scalar T. GRADIENT OF A SCALAR (Cont’d) The vector inside square brackets defines the change of temperature corresponding to a vector change in position. ![]() GRADIENT OF A SCALAR (Cont’d) But, So, previous equation can be rewritten as: GRADIENT OF A SCALAR (Cont’d) The differential distances are the components of the differential distance vector : However, from differential calculus, the differential temperature: VECTOR CALCULUS 1.10 GRADIENT OF A SCALAR 1.11 DIVERGENCE OF A VECTOR 1.12 DIVERGENCE THEOREM 1.13 CURL OF A VECTOR 1.14 STOKES’S THEOREM 1.15 LAPLACIAN OF A SCALARġ.10 GRADIENT OF A SCALAR Suppose is the temperature at, and is the temperature at as shown.
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |