For a gentle appetiser, see Spivak's Calculus on Manifolds. It introduces exterior algebra as a means to and end in doing computations on (embedded) manifolds.

For a gentle appetiser, see Spivak's Calculus on Manifolds. It introduces exterior algebra as a means to and end in doing computations on (embedded) manifolds. At first, the definitions seem opaque and the formalism clunky. At the end, Spivak recovers Stokes' theorem as a computation. This was a breath of fresh air for me after I took a warped version of Calc III.