![]() It is $\mathbf G \mathbf I$ that is the 2-form $\omega$, and it is $\nabla \wedge$ that truly represents, for all objects, the exterior derivative. I submit to you, for reasons too complex to get into, that $(\nabla \cdot \mathbf G) \mathbf I = \nabla \wedge $ using the underlying rules of Clifford algebra that geometric calculus is built on top of. The 3-vector $\mathbf I$ turns directions to orthogonal planes and points to volumes. $$\int_M (\nabla \cdot \mathbf G) (\mathbf I \, dV)$$ Rather, the "need" to do so follows from algebra. We can meaningfully integrate vector field divergences this way without arbitrarily converting them to two-forms. Since all 3-vectors in 3d are scalar multiples of each other, we call this simply $\mathbf I \, dV$. So instead of $dV$, we integrate over $d\mathbf V$, a 3-vector measure. Chapter 2 introduces tangent vec-tors and vector elds in IRn using the standard two approaches with curves and derivations. It enforces that when we integrate over geometric regions (surfaces, volumes, and so on), the integration measure itself not only contains magnitude formation (the area of the surface, the size of the volume) but direction information, too. Chapter 1 reviews some basic facts about smooth functions from IRn to IRm, as well as the basic facts about vector spaces, basis, and algebras. What's actually going on here can be explained in a different formalism called geometric calculus. I have background of linear algebra and advanced calculus. At starting point, I am not looking for a comprehensive book (may be Spivak's Comprehensive Introduction to Differential Geometry series). You have to take vector fields and turn them into two-forms to take meaningful divergences you have to convert them to one-forms to take meaningful curls (though this is much less peculiar). I am an engineering major and looking for a straightforward, easy to understand basic book on differential geometry to get started. ![]() You have to take all the stuff you're familiar with and turn it upside down. But this is the big problem with learning differential forms when coming from vector calculus. ![]() A differential k-form on an n-manifold can be visualised as a "density" of (n - k) submanifolds.
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