1. VECTOR CALCULUS - Line Integrals,Curl & Gradient
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. Define Work or Energy Change:
LINE INTEGRALS
EXAMPLE: GRAVITY
Define Work or Energy Change: 1 2
(0,0,h)
(0,0,0)
(0,h,0)
Along
(dx & dy = 0) 1

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

5. Note: For the first integral, DON’T use
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
NOTATION:
Implies a CLOSED LOOP Integral

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. ROTATION or CURL NOTATION: is NOT a CROSS-PRODUCT
Result of this operation is a VECTOR
Note: We will not go through the derivation of the connection between
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:

10. ROTATION or CURL 4 other terms include: two Af terms and two Ar terms
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 IF
, then
and