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Fields and Waves

Fields and Waves. Lesson 2.1. VECTORS and VECTOR CALCULUS. VECTORS. Today’s Class will focus on:. vectors - description in 3 coordinate systems. vector operations - DOT & CROSS PRODUCT. vector calculus - AREA and VOLUME INTEGRALS. z. y. x. VECTOR NOTATION. VECTOR NOTATION:.

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Fields and Waves

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  1. Fields and Waves Lesson 2.1 VECTORS and VECTOR CALCULUS Darryl Michael/GE CRD

  2. VECTORS Today’s Class will focus on: • vectors - description in 3 coordinate systems • vector operations - DOT & CROSS PRODUCT • vector calculus - AREA and VOLUME INTEGRALS

  3. z y x VECTOR NOTATION VECTOR NOTATION: Rectangular or Cartesian Coordinate System Dot Product (SCALAR) Cross Product (VECTOR) Magnitude of vector

  4. Choice is based on symmetry of problem VECTOR REPRESENTATION 3 PRIMARY COORDINATE SYSTEMS: • RECTANGULAR • CYLINDRICAL • SPHERICAL Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL

  5. z r P z x f y VECTOR REPRESENTATION: CYLINDRICAL COORDINATES UNIT VECTORS: Cylindrical representation uses: r ,f , z Dot Product (SCALAR)

  6. z P q r x f y VECTOR REPRESENTATION: SPHERICAL COORDINATES UNIT VECTORS: Spherical representation uses: r ,q , f Dot Product (SCALAR)

  7. z y x VECTOR REPRESENTATION: UNIT VECTORS Rectangular Coordinate System Unit Vector Representation for Rectangular Coordinate System The Unit Vectors imply : Points in the direction of increasing x Points in the direction of increasing y Points in the direction of increasing z

  8. z r P z f x y VECTOR REPRESENTATION: UNIT VECTORS Cylindrical Coordinate System The Unit Vectors imply : Points in the direction of increasing r Points in the direction of increasing j Points in the direction of increasing z

  9. z P q r f y x VECTOR REPRESENTATION: UNIT VECTORS Spherical Coordinate System The Unit Vectors imply : Points in the direction of increasing r Points in the direction of increasing q Points in the direction of increasing j

  10. Do Problem 1 VECTOR REPRESENTATION: UNIT VECTORS Summary RECTANGULAR Coordinate Systems CYLINDRICAL Coordinate Systems SPHERICAL Coordinate Systems NOTE THE ORDER! r,f, z r,q ,f Note: We do not emphasize transformations between coordinate systems

  11. Unit is in “meters” y df r x METRIC COEFFICIENTS 1. Rectangular Coordinates: When you move a small amount in x-direction, the distance is dx In a similar fashion, you generate dy and dz ( dx, dy, dz ) Generate: 2. Cylindrical Coordinates: Differential Distances: Distance = r df ( dr, rdf, dz )

  12. z P q r r sinq f y x y df x METRIC COEFFICIENTS 3. Spherical Coordinates: Differential Distances: Distance = r sinq df ( dr, rdq, r sinq df )

  13. METRIC COEFFICIENTS Representation of differential length dl in coordinate systems: rectangular cylindrical spherical

  14. dy dx y 6 2 3 7 x AREA INTEGRALS • integration over 2 “delta” distances Example: AREA = = 16 Note that: z = constant In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or f = constant or q = constant et c….

  15. PROBLEM 2 PROBLEM 2A What is constant? How is the integration performed? What are the differentials? Representation of differential surface element: Vector is NORMAL to surface

  16. DIFFERENTIALS FOR INTEGRALS Example of Line differentials or or Example of Surface differentials or Example of Volume differentials

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