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Fields and Waves

Fields and Waves. Lesson 2.2. VECTOR CALCULUS - Line Integrals,Curl & Gradient. DIFFERENTIAL LENGTHS. Representation of differential length dl in coordinate systems:. rectangular. cylindrical. spherical. (0,0,h). 1. 1. 2. (0,h,0). (0,0,0). LINE INTEGRALS. EXAMPLE: GRAVITY.

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Fields and Waves

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  1. Fields and Waves Lesson 2.2 VECTOR CALCULUS - Line Integrals,Curl & Gradient Darryl Michael/GE CRD

  2. DIFFERENTIAL LENGTHS Representation of differential length dl in coordinate systems: rectangular cylindrical spherical

  3. (0,0,h) 1 1 2 (0,h,0) (0,0,0) LINE INTEGRALS EXAMPLE: GRAVITY Define Work or Energy Change: (dx & dy = 0) Along

  4. 2 LINE INTEGRALS (dx & dy = 0) Along Vectors are perpendicular to each other Final Integration:

  5. LINE INTEGRALS Note: For the first integral, DON’T use Negative Sign comes in through integration limits For PROBLEM 1a - use Cylindrical Coordinates: Differential Line Element Implies a CLOSED LOOP Integral NOTATION:

  6. LINE INTEGRALS - ROTATION or CURL Measures Rotation or Curl For example in Fluid Flow: Means ROTATION or “EDDY CURRENTS” Integral is performed over a large scale (global) However, , the “CURL of v” is a local or a POINT measurement of the same property

  7. Note: We will not go through the derivation of the connection between ROTATION or CURL NOTATION: is NOT a CROSS-PRODUCT Result of this operation is a VECTOR More important that you understand how to apply formulations correctly To calculate CURL, use formulas in the TEXT

  8. ROTATION or CURL Example: SPHERICAL COORDINATES We quickly note that:

  9. ROTATION or CURL , has 6 terms - we need to evaluate each separately Start with last two terms that have f - dependence: 0

  10. ROTATION or CURL 4 other terms include: two Af terms and two Ar terms 0 Example: , because Aq has NO PHI DEPENDENCE THUS, Do Problem 1b and Problem 2

  11. GRADIENT GRADIENT measures CHANGE in a SCALAR FIELD • the result is a VECTOR pointing in the direction of increase For a Cartesian system: You will find that Do Problems 3 and 4 ALWAYS , then and IF

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