Gradient & Related CalculationsExample: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 vectorApplying 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 followingwhere i1= i, and i2= j, and dx1=dx, dx2=dy, and dx3=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(=∆st)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 y2xQ x y,()∴–2x–y–==∇Qiˆ∂∂x-----jˆ∂∂y-----kˆ∂∂z-----++Q()iˆ∂Q∂x-------jˆ∂Q∂y-------kˆ∂Q∂z-------++==∂Q∂x-------1, ∂Q∂y-------–1, and ∂Q∂z-------–0===∇Qiˆ–jˆ–=tˆiˆ1dx1ds-------iˆ2dx2ds-------iˆ3dx3ds-------++=xy12341432Q = 1Q = 0Q = -1Q = -2iˆ–jˆ–∇Q0r1r2Fig. 1∆s∆x∆y
has intentionally blurred sections.
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