Advanced Calculus: A Geometric View by James J. Callahan

By James J. Callahan

With a clean geometric method that comes with greater than 250 illustrations, this textbook units itself except all others in complicated calculus. along with the classical capstones--the swap of variables formulation, implicit and inverse functionality theorems, the fundamental theorems of Gauss and Stokes--the textual content treats different very important issues in differential research, resembling Morse's lemma and the Poincaré lemma. the information at the back of so much subject matters may be understood with simply or 3 variables. This invitations geometric visualization; the booklet contains glossy computational instruments to offer visualization actual strength. utilizing second and 3D pictures, the booklet deals new insights into basic parts of the calculus of differentiable maps, equivalent to the function of the by-product because the neighborhood linear approximation to a map and its position within the swap of variables formulation for a number of integrals. The geometric subject keeps with an research of the actual which means of the divergence and the curl at a degree of element no longer present in different complicated calculus books. complicated Calculus: a geometrical View is a textbook for undergraduates and graduate scholars in arithmetic, the actual sciences, and economics. must haves are an creation to linear algebra and multivariable calculus. there's sufficient fabric for a year-long path on complex calculus and for a number of semester courses--including issues in geometry. It avoids duplicating the cloth of actual research. The measured speed of the booklet, with its wide examples and illustrations, make it specially compatible for self sufficient study.

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This fact is established in the exercises. In what follows, {e1 , e2 , . . , en } is the standard basis for Rn . As you can see, the definitions relate the orientation and volume of an n-dimensional parallelepiped to the determinant of a certain n × n matrix. We review the definition of an n × n determinant in the exercises. 3 An ordered set {v1 , v2 , . . , vn } has positive orientation if detV > 0, where V : Rn → Rn is the linear map defined by the conditions V (ei ) = vi and i = 1, 2, . .

Of course we associate eigenvalues with stretches; in this example the shear is combined with a uniform dilation whose magnitude is given by the single eigenvalue 2. Shears M8 = x 4 −2 2 0 A shear can take a less recognizable form, as in the following example. M8 : x = 4u − 2v, y = 2u, M8 = v 4 −2 . 2 0 y M8 u x Because tr M8 = det M8 = 4, M8 has the single eigenvalue λ = 2. There is an eigenvector in only one direction. 1 Maps from R2 to R2 39 G: u = u + v, v = u, converts M8 into M7 : G−1 M8 G = M7 .

Exercises 25 We recall here some ideas introduced in the first course in multivariable calculus. Suppose that either (a) C F · dx = 0 for all closed paths C, or (b) C1 F · dx = C2 F · dx for every pair of oriented paths C1 and C2 that start at the same point A and end at the same point B. Then the vector field F is said to be conservative if (a) is true, and path-independent if (b) is true. It can be shown that a continuous and everywheredefined vector field is conservative if and only if it is path-independent.

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