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The gradient of  is de ned by   @ @ @ @ @ @ grad  r i j k  i j k @x @y @z @x @y @z @ @ @ i j k @x @y @z 2. Divergence. The divergence of A is de ned by   @ @ @ div A r A i j k A1 i A2 j A3 k @x @y @z @A @A @A 1 2 3 @x @y @z 3. Curl. The curl of A is de ned by   @ @ @ curl A r A i j k A1 i A2 j A3 k @x @y @z    i j k     @ @ @     @x @y @z    A A A  1 2 3        @  @ @   @ @  @        i @y @z  j @x @z  k @x @y         A A   A1 A2  A A  2 3 1 2       @A3 @A2 @A1 @A3 @A2 @A1 i j k @y @z @z @x @x @y
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Note that in the expansion of the determinant, the operators @=@x; @=@y; @=@z must precede A1 ; A2 ; A3 . In other words, r is a vector operator, not a vector. When employing it the laws of vector algebra either do not apply or at the very least must be validated. In particular, r A is a new vector obtained by the speci ed partial di erentiation on A, while A r is an operator waiting to act upon a vector or a scalar. FORMULAS INVOLVING r If the partial derivatives of A, B, U, and V are assumed to exist, then 1. r U V rU rV or grad U V grad u grad V 2. r A B r A r B or div A B div A div B 3. r A B r A r B or curl A B curl A curl B 4. r UA rU A U r A 5. r UA rU A U r A 6. r A B B r A A r B 7. r A B B r A B r A A r B A r B 8. r A B B r A A r B B r A A r B 9: r rU  r2 U  and r2  10. r rU 0. @2 U @2 U @2 U 2 2 is called the Laplacian of U @x2 @y @z @2 @2 @2 2 2 is called the Laplacian operator: 2 @x @y @z
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The curl of the gradient of U is zero.
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11. 12.
r r A 0. The divergence of the curl of A is zero. r r A r r A r2 A
VECTOR INTERPRETATION OF JACOBIANS, ORTHOGONAL CURVILINEAR COORDINATES The transformation equations x f u1 ; u2 ; u3 ; y g u1 ; u2 ; u3 ; z h u1 ; u2 ; u3 15 [where we assume that f ; g; h are continuous, have continuous partial derivatives, and have a singlevalued inverse] establish a one-to-one correspondence between points in an xyz and u1 u2 u3 rectangular coordinate system. In vector notation the transformation (17) can be written r xi yj zk f u1 ; u2 ; u3 i g u1 ; u2 ; u3 j h u1 ; u2 ; u3 k 16 A point P in Fig. 7-12 can then be de ned not only by rectangular coordinates x; y; z but by coordinates u1 ; u2 ; u3 as well. We call u1 ; u2 ; u3 the curvilinear coordinates of the point.
Fig. 7-12
If u2 and u3 are constant, then as u1 varies, r describes a curve which we call the u1 coordinate curve. Similarly, we de ne the u2 and u3 coordinate curves through P. From (16), we have dr @r @r @r du du du @u1 1 @u2 2 @u3 3 17
@r @r @r ; ; is a basis for the vector structure associated with the curvilinear @x @y @z system. If the curvilinear system is orthogonal, then so is this set; however, in general, the vectors are not unit vectors. he di erential form for arc length may be written The collection of vectors ds2 g11 du1 2 g22 du2 2 g33 du3 2 where g11 @r @r ; @x @x g22 @r @r ; @y @y g33 @r @r @z @z
The vector @r=@u1 is tangent to the u1 coordinate curve at P. If e1 is a unit vector at P in this direction, we can write @r=@u1 h1 e1 where h1 j@r=@u1 j. Similarly we can write @r=@u2 h2 e2 and @r=@u3 h3 e3 , where h2 j@r=@u2 j and h3 j@r=@u3 j respectively. Then (17) can be written
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