Evaluate F 2xyi x2j r ti t2j Identify P and Q Compute Ar

Evaluate F = 2xyi + x2j r = ti + t2j Identify P and Q. Compute Are they equal? If yes, continue. If not, do as in previous section. Find f such that fx = P &nd; fv = Q Check your answer, i.e. Show Find initial and final point on curve. Evaluate integral.

Solution

step 1) P= 2xy , Q= x^2 step 2) dQ/dx= 2x , and dP/dy =2x step 3) yes the are equal. step 4) fx= P=2xy ==> f=\\int 2xy dx = x^2y fy= Q= x^2 ==> f=\\int x^2 dy = x^2y step 5) F= (d/x i + d/dy j ) .f = (d/x i + d/dy j ) .(x^2y) = 2xyi+ x^2j step 6) r=ti + t^2 j ==> x= t and y= t^2, 0< t < 1 initial point (0,0) and final point (1,1) t=0 ==> x=0 and y=0 t=1 ==> x=1 and y=1 step 7) F= 2xyi +x^2 j , 0< t < 1 r=ti + t^2 j ==> x= t and y= t^2 ==> F(t) = 2t t^2 i +2t^2j = 2(t^3i +t^2 j) dr=dt i +2 t dt j F.dr = 2(t^3i +t^2 j).(dt i +2 t dt j ) = 2( t^3 +2t^3 ) dt \\int_{0}^{1} F. dr = \\int_{0}^{1} 2( t^3 +2t^3 ) dt = 3/2