Suppose we have the open surface S, the upper hemisphere (with no bottom): z = f (x, y) = /1 – x² – y². If the fluid velocity field is F (x, y, z) = (y, –x, z²), then evaluate the polar double integral that is equivalent to: Net Upward Flux = // (F î) dS = ||(-Mfz– Nfy+P) dA, S R where în is the unit upward normal to S, and the region R is the projection of S down onto the xy-plane.

Sketch how water curls down a sink, say, in clock-wise rotation. Draw the resulting vector of the curl-operator applied on this water flow.

Inside the wall of a house, an L-shaped section of hot-water pipe consists of three parts: a straight, horizontal piece h = 28.0 cm long, an elbow, and a straight vertical piece l = 152 cm long (figure below). A stud and a second-story floorboard hold the ends of this section of copper pipe stationary. Find the magnitude and direction of the displacement of the pipe elbow when the coordinate system with it's origin at the elbow, the +x-axis to the right in line with the horizontal water flow is turned on, raising the temperature of the pipe from 18.0°C to 43.6°C. (Assume pipe and the +y-axis going upward.) 0.6568 Your response is within 10% of the correct value. This may be due to roundoff error, or you could have a mistake in your calculation. Carry out all intermediate results to at least four-digit accuracy to minimize roundoff error. mm 79.57 magnitude direction ° (clockwise from the +x-axis) Need Help? Read It