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Contents. Ch. 8: DIFFERENTIAL OPERATORS I and II. Del Operator Gauss and Stokes Theorems Curvilinear Coordinates (Ch9). Del Operator. Del Operator. Definition. Gradient of a function (scalar) Definition: Example: force-potential energy relation. Curl of a vector field Definition:

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  1. Contents Ch. 8: DIFFERENTIAL OPERATORS I and II • Del Operator • Gauss and Stokes Theorems • Curvilinear Coordinates (Ch9)

  2. Del Operator Del Operator Definition • Gradient of a function (scalar) Definition: Example: force-potential energy relation • Curl of a vector field Definition: Example:

  3. Del Operator Del Operator (cont’d) • Divergence of a vector field Definition: Example: • Laplacian of a function Definition: Example:

  4. Del Operator Del Operator (cont’d)

  5. II Gauss and Stokes II. Gauss and Stokes theorems II.A Integrals: definition of circulation and flux • Definitions of Line integrals and circulation: • Theorem: when a vector field is the gradient of a scalar field , its circulation is zero

  6. II Gauss and Stokes II. Gauss and Stokes theorems II.A (continued) Flux

  7. II Gauss and Stokes II. Gauss and Stokes theorems 8.2-1 Compute the flux of the gravitational field of the Sun: through a sphere centered on the Sun and of radius r 8.2-2 Compute the flux of the electric field through the following a surfaces: • A square of side L=3, lying in a y-z plane located at x=4 • A cube of side L=2 centered on the origin (take the sides parallel to the rectangular axes x,y and z) 8.2-3 Compute the flux of the vector field through a square of with its corners at the points P1=(2,0,0); P2=(2,2,0); P3=(2,2,2); P4=(2,0,2). 8.2-4 Compute the circulation around a circle of radius 2r centered on the origin of 8.2-5 Compute the circulation around a circle of radius 2r centered on the origin of 8.2-6 Compute the circulation around a rectangular loop (p1=(0,0,0); p2=(0,3,0); p3=(0,3,1); p4=(0,0,1)) of the vector field

  8. II. Gauss and Stokes II. Gauss and Stokes theorems (cont’d) II.B Gauss’ Theorem Theorem statement: Proof:

  9. II. Gauss and Stokes II Gauss and Stokes theorems (cont’d) The idea is to cut up the finite volume into the elementary cubes we considered on the previous slide.

  10. II. Gauss and Stokes II Gauss and Stokes theorems (cont’d) Example: on a cube of side d centered at O

  11. II Gauss and Stokes II. Gauss and Stokes theorems (cont’d) 8.2-7

  12. II Gauss and Stokes II Gauss and Stokes theorems (cont’d) II. C Stokes’ Theorem Statement: Proof: for elementary loop in, say, x-y plane: E8.2-1 Do the proof for an elementary loop lying in the z-y plane, then for a loop lying in the z-x plane.

  13. II Gauss and Stokes II. Gauss and Stokes theorems (cont’d)

  14. II Gauss and Stokes II. Gauss and Stokes theorems (cont’d) 8.2-10 8.2-11

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