Vector line integrals, Multivariable Calculus

We look at the derivation for vector line integrals, including three integral expressions we may encounter. Two examples.

This is Multivariable Calculus Unit 6 Lecture 7.

#lineintegral #iitjammathematics #calculus3

[eswyatt]

The last diagram in example 2 shows vectors on the inside of the bending path being shorter than those on the outside of the path. If a vector field's change in direction is necessary for the line integral through a closed path not to be zero, which intuitively seems correct, then the vectors inside the bend must be shorter than the ones outside the bend. In other words, there must be curl!! This foreshadows Stokes' theorem, an insight I didn't have before.