MET4305_02 - Gradient & Related Calculations Example: We...

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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 Qxy , () –2 x y == Qi 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 = ˆ j ˆ = t ˆ i ˆ 1 dx 1 ds i ˆ 2 2 i ˆ 3 3 = 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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dx and dy depend on ds only - hence the total derivatives. We can ‘estimate’ the slope using a finite value of s (and corresponding x, y) as shown in Fig. 1, we have Thus we have Note that because we are dealing with a straight line segment (i.e. constant slope) the approximation of the slope for relatively large x and y is ok (in fact it is exact here)! How- ever, you can’t generally do this because slopes for most curves can vary quite a bit over short distances (the slope is valid at a single point - i.e. in the limit as s approaches zero). Try doing this for a parabola! c.)What is the magnitude of the gradient of Q?
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This note was uploaded on 02/11/2012 for the course MET 4305 taught by Professor Lazarus during the Fall '09 term at FIT.

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MET4305_02 - Gradient & Related Calculations Example: We...

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