(1)Section 16.1 Ex11

Section 16.1 Ex11.

F (x, y) = (y, x) corresponds to graph (2) Ex12.

F (x, y) = (1, sin y) corresponds to graph (4) Ex13.

F (x, y) = (x − 2, x + 1) corresponds to graph (1) Ex14.

F (x, y) = (y,_x¹) corresponds to graph (3) Ex26.

f (x, y) =√

x²+ y² ⇒ ∇f (x, y) = (√ ^x

x²+y²,√ ^y

x²+y²) Ex35.

(a) We might guess that the ow lines have equations y = ^C_x.

(b)If x = x(t) and y = y(t) are parametric equations of a ow line , then the velocity vector of the ow line at the point (x, y) is x⁰(t)i + y⁰(t)j. Since the velocity vectors coincide with the vectors in the vector eld , we have x⁰(t)i + y⁰(t)j = xi − yj ⇒ dx/dt = x, dy/dt = −y. To solve these dierential equations , we know dx/dt = x ⇒ dx/x = dt ⇒ ln |x| = t + C ⇒ x =

±e^t+C = Ae^t for some constant A , and dy/dt = −y ⇒ dy/y = −dt ⇒ ln |y| = −t + K ⇒ y = ±e^−t+K = Be^−t for some constant B. Therefore xy = Ae^tBe^−t = AB =constant. If the ow lines pass through (1, 1) then (1)(1) =constant= 1 ⇒ xy = 1 ⇒ y = 1/x, x > 0

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