32 Find the value of the line integral d c F r where Fx y

Solution

You are trying to find the line integral over the two different paths. First, you can see that F(x,y,z) is a conservative field, meaning that it is the gradient of a function (namely, f=5xyz). Therefore, we can use the fundamental theorem of line integrals and disregard the path, we only need the endpoints. For the first path, r_start=<0,10,0>, r_end=<100,10,100> f(r_end)-f(r_start)=5*100*100*10-0=500000 For the second path, r_start=<0,10,0>, r_end=<10,10,10> f(r_end)-f(r_start)=5*10*10*10-0=5000 i)500000 ii) 5000 This is the answer I get, but it doesn\'t seem to be an answer choice. It may be due to the question being spotty at best in terms of readability and I solved the wrong problem.