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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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slide1

Fields and Waves

Lesson 2.2

VECTOR CALCULUS - Line Integrals,Curl & Gradient

Darryl Michael/GE CRD

slide2

DIFFERENTIAL LENGTHS

Representation of differential length dl in coordinate systems:

rectangular

cylindrical

spherical

slide3

(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

slide4

2

LINE INTEGRALS

(dx & dy = 0)

Along

Vectors are perpendicular to each other

Final Integration:

slide5

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:

slide6

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

slide7

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

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ROTATION or CURL

Example: SPHERICAL COORDINATES

We quickly note that:

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ROTATION or CURL

, has 6 terms - we need to evaluate each separately

Start with last two terms that have f - dependence:

0

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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

slide11

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