BASIC ELEMENTS OF VECTOR CALCULUS

In the following, S is a scalar function of x,y,z: S(x,y,z) in the following, A and C are vector functions of x,y,z:

A = Ax(x,y,z) x + Ay(x,y,z) y + Az(x,y,z) z Where x, y, and z are unit vectors in the x, y, and z directions respectively [rectangular], and r, , and  are unit vectors in the r,, and  directions [polar, cylindrical, and spherical].

 = del operator: in rectangular form it is

 = ([ ]/x) x + ([ ]/y) y + ([ ]/z) z

gradient: S = (S/x) x + (S/y) y + (S/z) z (a vector!)

gradient shows the change in S over space and is in the direction of that greatest change, and hence is a vector.

divergence: A = (Ax/x) + (Ay/y) + (Az/z) (a scalar!)

idea of divergence is best indicated through divergence theorem:

 A dVolume = closed surface An dSurface surface enclosed volume

Use in E&M: One of Maxwell’s Equations is Gauss’ Law for Magnetic Fields:  B  dA = 0 closed area Using the Divergence Theory, this can be written as   B = 0. Another of Maxwell’s Equations is Gauss’ Law for Electric Fields:  E  dA = Q /  closed area enclosed o . In empty space (where Qenclosed = 0) this can be written as   E = 0.

curl: A = [(Az/y)-(Ay/z)]x + [(Ax/z)-(Az/x)]y + [(Ay/x-Ax/y)]z

idea of curl is best indicated through Stokes theorem:

 (A)n dArea = closed path Adlength length encircles area

Use in E&M: One of Maxwell’s Equations is Faraday’s Law:  E  dL = V = d/dt   B  dA ]. closed loop Using Stokes’ Theorem, this can then be written as   E = dB/dt . Another of Maxwell’s Eq.s is Ampere’s Law:  B  dL =  I +   d/dt [  E  dA] closed loop o encircled o o In empty space (where Iencircled = 0) this can be written as   B =   dE/dt o o

IDENTITIES (A+C) = A + C (scalar)

(SA) = (S)A + S(A) (scalar)

(S) = ²S = (²S/x² + ²S/y² + ²S/z²) (scalar called Laplacian)

(A+C) = (A) + (C) (vector)

(SA) = (S)A + S(A) (vector)

(AC) = C(A) - A(C) (scalar)

(AC) = (C)A - (A)C + (C)A - (A)C (vector)

(AC) = (C)A + (A)C + C(A) + A(C) (vector)

(A) = (A) - ²A (vector)

(S) = 0 (vector)

(A) = 0 (scalar)

In spherical coordinates:

 = r ( /r) +  (1/r) ( /) +  (1/[r sin]) ( /)

(S) = ²S = (1/r²) (/r)(r² S/r) + (1/[r²sin]) (/)(sin S/)

+ (1/[r²sin²]) (²S/²)

In cylindrical coordinates:

 =  ( /) +  (1/) ( /) + z ( /z)

(S) = ²S = (1/) (/)( S/) + (1/²) (²S/²) + (²S/z²)

Homework Problem #17: Calculate the curl of A, A, in cylindrical coordinates. To do this, start with  in cylindrical coordinates (see above), then express A in cylindrical coordinates: A = Ar r + Af f + Az z, where all three Ai’s are functions of r, f, and z; and recall that both r and f are functions of f. HINT: as an example, see previous section where A was calculated in spherical coordinates. HINT: cylindrical coordinates are simply 2-D polar with z added, where r acts like r and f acts like θ. WARNING: since r and f are functions of f, you can’t use the matrix way of calculating the cross product since that assumes the unit vectors are constant (as they are in rectangular form).

Interesting theorems based on the last two identities:

1.  (S) = 0

If   A = 0 , then A can be expressed as the gradient of some scalar field, S: A = S . In E&M this is used to relate an Electric field vector, E, to a scalar potential (voltage).

2.  (A) = 0 If   C = 0 , then C can be expressed as the curl of some other vector field, A, C = (A) .

In E&M this is used to relate a Magnetic field vector, B, to a vector potential, A.