mathvects

The magnitude of the result can be calculated from

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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 Solution: ^^k ij^ ~ B = 1 0 0 = 0^ + 0^ + 2k ~ A ij^ 120 Gradient and Divergence The Gradient 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 i j^ Gradients of certain functions are important throughout engineering. Some examples are (1) The heat ux transferred through a surface is ~ = ;krT q 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 ~ V =r where is a scalar function. For ~ V (x y) = x^ ; y^ ij Then @ =x @x and so integrating: 2 (x y) = x + C (y) 2 where C (y) at this point is an arbitrary function of y. Also @ = ;y @y and thus 2 (x y) = ; y2 + D(x) where D(x) is another function of x. Comparing the above equations for , we...
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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.

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