Notes 3 Review of Vector Calculus

Transcription

1 ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 A ˆ Notes 3 Review of Vector Calculus y ya ˆ y x xa V = x y ˆ x Adapted from notes by Prof. Stuart A. Long 1

2 Overview Here we present a brief overview of vector calculus. A much more thorough discussion of vector calculus may be found in the class notes for ECE 3318: Notes 13: Divergence Notes 17: Curl Notes 19: Gradient and Laplacian Please also see the textbooks and the following supplementary books (on reserve in the Library): H. M. Schey, Div, Grad, Curl, and All That: an Informal Text on Vector Calculus, 2 nd Ed., W. W. Norton and Company, M. R. Spiegel, Schaum s Outline on Vector Analysis, McGraw-Hill,

3 Del Operator xˆ + yˆ + ˆ x y This is an operator. Gradient φ φ φ φ = xˆ + yˆ + ˆ x y (Vector) Laplacian φ = ( ) φ = + + φ x y 2 (Scalar) 3

4 Del Operator (cont.) = xˆ + yˆ + ˆ x y Vector A: A= A xˆ+ A yˆ+ A ˆ x y Divergence A A A x y x y A = + + (Scalar) Curl A A y A A x A y Ax A xˆ yˆ = + + ˆ y x x y (Vector) Note: Results for cylindrical and spherical coordinates are given at the back of your books. 4

5 Vector Identities Two fundamental ero identities: A = ( ) 0 φ = ( ) 0 5

6 Vector Identities (cont.) Another useful identity: ( A B) = B ( A) A ( B) This will be useful in the derivation of the Poynting theorem. 6

7 Vector Laplacian The vector Laplacian of a vector function is a vector function. 2 A ( A) ( A) The vector Laplacian is very useful for deriving the vector Helmholt equation (the fundamental differential equation that the electric and magnetic fields obey). 7

8 Vector Laplacian (cont.) In rectangular coordinates, the vector Laplacian has a very nice property: ( 2 ) ( 2 ) ( 2 ) x y A= xˆ A yˆ A ˆ A This identity is a key property that will help us reduce the vector Helmholt equation to the scalar Helmholt equation, which the components of the fields satisfy. 8

9 Gradient φ φ φ φ =xˆ + yˆ + ˆ x y φ φ φ d φ = dx + dy + d x y (from calculus) φ φ φ = φ dr = xˆ + yˆ + ˆ xˆ( dx) + yˆ( dy) + ˆ( d) x y dφ φ dr dr = = φ d d d d φ = d φ ( ) φ d r dr r + dr The gradient vector tells us the direction of maximum change in a function. 9

10 Gradient (cont.) Rectangular Φ Φ Φ Φ = xˆ + yˆ + ˆ x y Cylindrical ˆ 1 Φ = ˆ ρ Φ + φ Φ + ẑ Φ ρ ρ φ Spherical ˆ1 ˆ 1 Φ = rˆ Φ + θ Φ + φ Φ r r θ rsinθ φ 10

11 Divergence A ˆ 1 A = Lim A nˆ ds V 0 V S x ˆ x ya ˆ y xa V = x y y The divergence measures the rate at which flux of the vector function emanates from a region of space. Divergence > 0: source of flux Divergence < 0: sink of flux Please see the books or the ECE 3318 class notes for a derivation of this property. 11

12 Divergence (cont.) Rectangular: A A A A x y x y = + + Cylindrical: 1 1 Aφ A A= ( ρ Aρ ) + + ρ ρ ρ φ Spherical: Aφ A= ( r Ar ) + ( Aθ sinθ ) + sinθ θ sinθ φ 2 r r r r 12

13 C y, 0 y ˆx "right-hand rule for C" S = y x y Curl 1 A x = A dr y ( ) ˆ Lim y, 0 A component of the curl tells us the rotation of the vector function about that axis. A = velocity vector ( A dr gives force on vanes) C River C ( A) x< ˆ 0 Paddle wheel y Please see the books or the ECE 3318 class notes for a derivation of this property. 13

14 Curl (cont.) Curl is calculated here x S x S C x C S y C y y 1 A xˆ Lim A dr Sx 0 S ( ) 1 A yˆ Lim A dr Sx 0 S ( ) 1 A ˆ Lim A dr Sx 0 S ( ) x C y C C Note: The paths are defined according to the right-hand rule. x y Note: The paths are all centered at the point of interest (a separation between them is shown for clarity). 14

15 Rectangular Curl (cont.) A Ay A A x A y A x A= xˆ + yˆ + ˆ y x x y Cylindrical ( ρ Aφ ) 1 A A φ ˆ Aρ A 1 A ρ A= ˆ ρ + φ + ˆ ρ φ ρ ρ ρ φ Spherical ( A sinθ ) ( ra ) ( ra ) 1 φ A 1 θ 1 A r φ ˆ 1 θ A r A= rˆ + θ + φ r sinθ θ φ r sinθ φ r r r θ 15

16 Divergence Theorem ˆn V S ˆn = outward normal V A dv = A nˆ ds S A = arbitrary vector function 16

17 Stokes s Theorem ˆn S (open) C (closed) The unit normal is chosen from a right-hand rule according to the direction along C. (An outward normal corresponds to a counter clockwise path.) S A n ds = A dr ( ) ˆ C 17