Fields and Waves

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