Unformatted text preview: esult can be calculated from the formula: ~~
j A B j=j A jj B j sin where is the angle between the vectors as in Figure 3. Note that
A B = ;B A ~
Example 4: Given: A = (1 0 0), B = (1 2 0). Find: A B
~ B = 1 0 0 = 0^ + 0^ + 2k
120 Gradient and Divergence
4 The gradient of a scalar function is the vector
r = @ ^+ @ ^ + @ k
@x i @y j @z
Example 5. Given = 2xy . Find: the gradient.
Solution: Substituting from the de nition, r = 2y^ + 2x^ + 0k
j^ Gradients of certain functions are important throughout engineering. Some
(1) The heat ux transferred through a surface is ~ = ;krT
where k is the thermal conductivity.
q2 y q x
q 3 z Figure 4: Sketch of heat ux through a cube.
5 1 (2) The net pressure force per unit volume on a uid element is
Fp = ;rp
where p is the pressure.
(3) For some velocity elds it is possible to write
where is a scalar function. For
V (x y) = x^ ; y^
and so integrating:
(x y) = x + C (y)
where C (y) at this point is an arbitrary function of y. Also
@ = ;y
(x y) = ; y2 + D(x)
where D(x) is another function of x. Comparing the above equations for ,
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This note was uploaded on 02/17/2013 for the course AME 331 taught by Professor Zohar during the Spring '08 term at Arizona.
- Spring '08