MET4305_02 - Gradient Related Calculations Example We are...

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Gradient & Related Calculations Example : We are out recording measurements of some scalar quantity Q and after many observations of this quantity (in both space and time) we arrive at the following family of curves (for different values of Q): a.) Calculate the gradient of Q . From our analytic expression above we have: Thus the gradient of Q is given by the Cartesian vector Applying graphical vector analysis, we can plot the gradient (it is -1 in the x-direction, and -1 in the y direction) vector (does it matter where I put it, i.e. have students tell me that the gradient is uniform, therefore I can plop it down anywhere on Fig. 1 above - of course this won’t always be the case!). b.)What is the tangent vector to the isolines? We can apply the following where i 1 = i , and i 2 = j , and dx 1 =dx, dx 2 =dy, and dx 3 =dz=0 (xy plane only). So we need to calcu- late the slopes dx/ds, dy/ds. These slopes are just the limit as the difference between the two posi- tion vectors r (= s t )approaches zero (recall that the slope is valid at a ‘pt’). Imagine the two points on the isoline - green pts in Fig.1, approaching one another and as they do, the triangle that constitutes the components of the vector drawn along the isoline, shrinks in area. Note that both y 2 x Q x y , ( ) 2 x y = = Q i ˆ x ----- j ˆ y ----- k ˆ z ----- + + Q ( ) i ˆ Q x ------- j ˆ Q y ------- k ˆ Q z ------- + + = = Q x ------- 1, Q y ------- 1, and Q z ------- 0 = = = Q i ˆ j ˆ = t ˆ i ˆ 1 dx 1 ds ------- i ˆ 2 dx 2 ds ------- i ˆ 3 dx 3 ds ------- + + = x y 1 2 3 4 1 4 3 2 Q = 1 Q = 0 Q = -1 Q = -2 i ˆ j ˆ Q 0 r 1 r 2 Fig. 1 s x y
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