By James J. Callahan

With a clean geometric method that includes greater than 250 illustrations, this textbook units itself except all others in complicated calculus. in addition to the classical capstones--the swap of variables formulation, implicit and inverse functionality theorems, the crucial theorems of Gauss and Stokes--the textual content treats different very important issues in differential research, akin to Morse's lemma and the Poincaré lemma. the information in the back of so much subject matters should be understood with simply or 3 variables. This invitations geometric visualization; the ebook accommodates sleek computational instruments to offer visualization actual strength. utilizing second and 3D photographs, the publication deals new insights into primary 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 function within the swap of variables formulation for a number of integrals. The geometric subject matter keeps with an research of the actual that means of the divergence and the curl at a degree of element now not present in different complex calculus books. complex Calculus: a geometrical View is a textbook for undergraduates and graduate scholars in arithmetic, the actual sciences, and economics. necessities are an advent to linear algebra and multivariable calculus. there's adequate fabric for a year-long direction on complicated calculus and for a number of semester courses--including subject matters in geometry. It avoids duplicating the cloth of actual research. The measured speed of the ebook, with its large examples and illustrations, make it particularly appropriate for self sufficient examine.

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**Sample text**

In the new coordinates, the map X = M4U takes the form X = G−1 X = G−1 M4 U = G−1 M4 GU = M4 U. For us, the object of this string of equalities is the conclusion M4 = G−1 M4 G, which leads, finally, to the following definition. 1 Suppose A and B are n × n matrices; then we say that B is equivalent to A if there is an invertible matrix G for which B = G−1 AG. What we have just shown about M4 and M4 implies that if B is equivalent to A, then there is a basis of Rn on which A acts in the same way that B acts on the standard basis of Rn .

The work done by a force) as the new kind of scalar integral over that path without its orientation. 5), let us parametrize the oriented curve C by arc length: x = y(s), 0 ≤ s ≤ L. Then, for the vector function F(x), C F · dx = b a F(y(s)) · y′ (s) ds = L 0 F(y(s)) · t(s) ds, where t(s) = y′ (s) is the unit tangent vector at the point y(s) on C; its direction indicates the orientation of C. But the last integral is a way to evaluate the scalar function F · t over the unoriented curve C; in other words, we have C F · dx = C F · t ds.

Notice, however, that the diagonals of the square grid are invariant; they are shown dotted in the figure. The image of a vector that lies on the diagonals is just a multiple of itself: 12 21 1 3 1 = =3 , 1 3 1 12 21 −1 1 −1 = = −1 . 1 −1 1 Specifically, the diagonal in the first and third quadrants is stretched by the factor 3, whereas the diagonal in the second and fourth is simply flipped, with no change in length. The presence of a negative multiplier suggests that orientation is reversed, and that is confirmed by the clockwise circuit in the image.