the sky is green, the grass is red - by laura gao

Parameterize the curve. Plug it into the vector field and dot it with the derivative of the parametrization. Finally, take the definite integral over the bounds of your parameter. That's how you compute a line integral. On Friday, I did this process, like, ten times. I understand the line integral now, right? I can compute it for any parametrizeable 3D-and-probably-higher curve. Then, that night, I tried to prove something that involved line integrals. (If the line integral of any closed loop = 0, then vector field has to be a gradient vector field.) And it turns out, I didn't understand line integrals at all. Why is it a dot product? What does it mean to dot the vector with the differential unit? Are you summing up all the vectors along the curve? The components of the vectors in the direction of the curve? My class started off heavily proof-based. The first pset was just proving properties about real numbers, like “if a ≠ 0 and b ≠ 0, prove ab ≠ 0.” All the psets were proof-based thr

Parameterize the curve. Plug it into the vector field and dot it with the derivative of the parametrization. Finally, take the definite integral over the bounds of your parameter. That's how you compute a line integral. On Friday, I did this process, like, ten times. I understand the line integral now, right? I can compute it for any parametrizeable 3D-and-probably-higher curve. Then, that night, I tried to prove something that involved line integrals. (If the line integral of any closed loop = 0, then vector field has to be a gradient vector field.) And it turns out, I didn't understand line