Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
This little book is especially concerned with those portions of OCOadvanced calculusOCO in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approa"
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A C R Ak(V basis boundary bounded function calculus called chain rule Chapter classical theorems closed rectangle closed set compact set consider continuously differentiable coordinate system define F defined by f(x,y definition denoted Df(a Dif(a Dif(x,y differentiable function div F Divergence Theorem dz A dx equation fc-cube fc-dimensional manifold fc-form fc-tensor Figure finite number Fubini's theorem function g G Rn Hence Hint induced orientation inner product integrable interior intersects Jordan-measurable l)-form least upper bound Lemma linear transformation manifold-with-boundary mathematics matrix measure Michael Spivak ms(f n-chain non-zero open cover open interval open rectangle open set containing orientation-preserving partial derivatives partition of unity Problem prove reader Rm is differentiable satisfies singular n-cubes sional manifold Stokes subrectangle subset suffices Suppose Theorem 2-2 theorem is true unique usual orientation vector field vector space volume element