Curl

26 When the vector operator V operates in a manner analogous to vector multiplication, the result is a vector, curl v called the curl of the vector field v (sometimes called the rotation).

curl v

ei

e2

e3

d

d

d

dxi

drX2

dx3

vi

v2

v3

dv2 \

f dvi

dx3 J

ei +

\dx3

dv3 dxi dv2 dxi dv\ dx2

Determine the curl of the following vector A = xz3i — 2x2yzj + 2yz4k at (1, —1,1). Solution:

= (2z4 + 2x2y)i + 3xz2j — 4xyzk = 3j + 4k at (1, — 1,1)

Mathematica solution is shown in Fig. 3.12.

<< Calculus'VectorAnalysis1 A = {xz*3, -2x*2yz, 2yzA4}; CurlOfA = Curl[A, Cartesian[x, y, z]] {2 z4 + 2 x2 y, 3 xZ, -4 xyz} << Graphics,PlotField3D1

PlotVectorField3D[ CurlOfA, {x, 0, 2}, {y, -2, 0}, {z, 0, 2}, Axes -> Automatic, AxesLabel-> {"x", "y", "z"}]

- Graphics3D -

CurlOfA

- Graphics3D -

CurlOfA

Figure 3.12: Mathematica Solution for the Curl of a Vector

3.6 Some useful Relations

-yian

0 0

Post a comment