Question

Let ∇f=−10xe−x2sin(3y)i+15e−x2cos(3y)j​. Find the change in f between (0,0) and (1,π/2) in two ways. (a) First, find the change by computing the line integral ∫C​∇f⋅dr, where C is a curve connecting (0,0) and (1,π/2). The simplest curve is the line segment joining these points. Parameterize it: with 0≤t≤1,r(t)=i+j​ So that ∫C​∇f⋅dr=∫01​ dt Note that this isn't a very pleasant integral to evaluate by hand (though we could easily find a numerical estimate for it). It's easier to find ∫C​∇f⋅dr as the sum ∫C1​​∇f⋅dr+∫C2​​∇f⋅dr, where C1​ is the line segment from (0,0) to (1,0) and C2​ is the line segment from (1,0) to (1,π/2). Calculate these integrals to find the change in f. ∫C1​​∇f⋅dr= ∫C2​​∇f⋅dr= So that the change in f=∫C​∇f⋅r=∫C1​​∇f⋅dr+∫C2​​∇f⋅dr= (b) By computing values of f. To do this, First find f(x,y)= Thus f(0,0)= and f(1,π/2)= and the change in f is

Answer 1

(a) The change in f between (0,0) and (1,π/2) can be computed using the line integral ∫C ∇f⋅dr, where C is the line segment connecting (0,0) and (1,π/2). Splitting the integral into two segments, from (0,0) to (1,0) and from (1,0) to (1,π/2), and evaluating each, the change in f is obtained as the sum of these integrals.

(b) Alternatively, we can compute the change in f by directly evaluating the function f(x,y) at the two given points (0,0) and (1,π/2). Subtracting the initial value from the final value gives the change in f.

Step-by-step explanation:

(a) The line integral ∫C ∇f⋅dr is calculated by parameterizing the curve C as a line segment from (0,0) to (1,π/2). The integral is split into two parts, each corresponding to a line segment. After parameterization and integration, the sum of these integrals provides the change in f .

(b) For the direct evaluation approach, we find f(x,y) as given. By substituting (0,0) and (1,π/2) into the function, we obtain the initial and final values of f. Subtracting these values gives the change in f .

Answer 2

To find the change in f between (0,0) and (1,π/2) in two ways: (a) Compute the line integral ∫C​∇f⋅dr, by breaking it down into two line segments, (b) Compute values of f(x,y) for (0,0) and (1,π/2) and calculate the change in f.

Step-by-step explanation:

To find the change in f between (0,0) and (1,π/2) using a line integral, we can break it down into two line integrals: one along the line segment from (0,0) to (1,0) and another along the line segment from (1,0) to (1,π/2). For the first line segment, with parameterization r(t) = ti, where 0 ≤ t ≤ 1, the line integral is given by ∫01∇f⋅dr = ∫01(-10te-t2sin(3t))dt. For the second line segment, with parameterization r(t) = i + t(π/2)i, where 0 ≤ t ≤ 1, the line integral is given by ∫01∇f⋅dr = ∫01(15e-1cos(3t2)dt). Evaluate these integrals separately to find the change in f.