The Del Operator Field Operations - EMPossible

1 9/5/2022. Electromagnetics: Electromagnetic Field Theory The Del Operator & Field Operations 1. Outline Scalar vs. Vector Fields The Del Operator . Gradient of a Scalar Field . Divergence of a Vector Field . Curl of a Vector Field . Laplacian Operation . Slide 2. 2. 1. 9/5/2022. Scalar vs. Vector Fields Slide 3. 3. Scalar Field Vs. Vector Field (2D). Scalar Field , Vector Field , . Magnitude only Magnitude Direction Slide 4. 4. 2. 9/5/2022. Scalar Field Vs. Vector Field (3D). Scalar Field , , Vector Field , , . Magnitude only Magnitude Direction Slide 5. 5. Isocontour Lines Isocontour lines trace the paths of equal value.

2 Closely space isocontours conveys that the function is varying rapidly. Slide 6. 6. 3. 9/5/2022. The Del Operator Slide 7. 7. The Del Operator . The del Operator is the vector differential Operator . It is sort of a 3D derivative. Even though is a Coordinate vector, it is never Del Operator System written as .. Cartesian a x a y a z x y z 1 . Cylindrical a a a z Derived from the Cartesian z equation using coordinate 1 1 . Spherical a r a a transformation. r r r sin . Slide 8. 8. 4. 9/5/2022. Summary of Vector Operations Operation Input Output Vector Addition & Subtraction Vectors Vector.

3 U V.. Dot Product U V Vectors Scalar . Cross Product U V Vectors Vector Gradient f Scalar Function Vector Function . Divergence U Vector Function Scalar Function . Curl U Vector Function Vector Function Scalar Laplacian V2. Scalar Function Scalar Function . Vector Laplacian 2U Vector Function Vector Function Slide 9. 9. Gradient of a Scalar Field Slide 10. 10. 5. 9/5/2022. Gradient of a Scalar Field (1 of 3). Start with a scalar Field , . Slide 11. 11. Gradient of a Scalar Field (2 of 3). Plot the gradient , on top of , . The background color is the original scalar Field .

4 Slide 12. 12. 6. 9/5/2022. Gradient of a Scalar Field (3 of 3). The gradient will always be perpendicular to the isocontour lines. Slide 13. 13. The Gradient . The gradient calculates how rapidly, and in what direction, a scalar function is increasing. Coordinate Gradient Operator System V V V. Cartesian V a x a y a z x y z V 1 V V. Cylindrical V a a a z z V 1 V 1 V. Spherical V a r a a . r r r sin . Slide 14. 14. 7. 9/5/2022. Algebra Rules for the Gradient U V U V Sum/Difference Rule UV U V V U Product Rule U V U U V. Quotient Rule V V2. V n nV n 1 V Power Rule Slide 15.

5 15. Properties of the Gradient 1. The gradient of a scalar function is a vector function. 2. The magnitude of V is the local maximum rate of change in V. 3. V points in the direction of maximum rate of change in V. 4. V at any point is perpendicular to the constant V surface that passes through that point. 5. V points toward increasing numbers in V. Slide 16. 16. 8. 9/5/2022. Divergence of a Vector Field Slide 17. 17. Divergence of a Vector Field (1 of 2). Start with the following vector Field , , . Observe how the Field seems to be converging to point in the upper left and diverging from a point in the low right.

6 Slide 18. 18. 9. 9/5/2022. Divergence of a Vector Field (2 of 2). The divergence is , , . Divergence measures the tendency of a vector Field to diverge from a point or converge to a point. Divergence is a scalar quantity and is shown as the colored regions. Blue indicates negative divergence (convergence) and red indicates positive divergence. Slide 19. 19. Divergence of a Vector Field (2D). Vector Field , Divergence , . Slide 20. 20. 10. 9/5/2022. Divergence . The divergence of a vector Field is a scalar Field that measures the tendency of a vector Field to diverge from a point or converge to a point.

7 Coordinate Gradient Operation System A Ay Az Cartesian A x . x y z 1 A 1 A Az Cylindrical A . z 1 r 2 Ar 1 A sin 1 A . Spherical A 2 . r r r sin r sin . Slide 21. 21. Algebra Rules for Divergence .. A B A B Distributive Rule Slide 22. 22. 11. 9/5/2022. Properties of Divergence . 1. The divergence of a vector function is a scalar function. 2. The divergence of a scalar Field does not make sense. 3. The original vector Field will point form larger to smaller numbers in the scalar Field calculated from the divergence. Slide 23. 23. Curl of a Vector Field Slide 24.

8 24. 12. 9/5/2022. Curl of a Vector Field (1 of 2). Start with the following vector Field , , . Observe how the Field seems to be rotating around two different axes. Slide 25. 25. Curl of a Vector Field (1 of 2). The curl is , , . Curl measures the tendency of the vector Field to circulate around an axis. Curl is a vector quantity and is shown as the blue arrows. The magnitude of the curl conveys the strength of the circulation. The direction of the curl is the axis of the circulation. Slide 26. 26. 13. 9/5/2022. Curl of a Vector Field ( 2D). Vector Field , Curl.

9 Y x z y x z Slide 27. 27. Curl The curl of a vector function is a vector function that quantifies the tendency of the vector Field to circulate around an axis. The magnitude of the curl conveys the strength of the circulation. The direction of the curl is the axis of the circulation. Coordinate Curl Operation System A Ay Ax Az Ay Ax . A z a x a y a z Cartesian y z z x x y . 1 Az A A Az 1 A A . Cylindrical A a a a z . z z . 1 A sin A 1 1 Ar rA 1 rA Ar . Spherical A a r a a . r sin r sin r r r . Slide 28. 28. 14. 9/5/2022. Algebra Rules for Curl .. A B A B Distributive Rule.

10 A B A B B A B A A B . Triple Product Slide 29. 29. Properties of Curl 1. The curl of a vector function is a vector function. 2. The curl of a scalar Field does not make sense.. 3. Curl follows the right hand rule. 4. The divergence of the curl of a vector Field is A. always zero. 0. 5. The curl of the gradient of a scalar Field is always zero. 0. 6. Note that . calculates the derivative of , whereas sets up . a derivative operation that is scaled by . A. Slide 30. 30. 15. 9/5/2022. Laplacian Operation Slide 31. 31. Scalar Laplacian . The scalar Laplacian is defined as the divergence of the gradient.