# Read e-book online A basic inequality for submanifolds in locally conformal PDF

By Tripathi M. M., Kim J., Kim S.

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Extra resources for A basic inequality for submanifolds in locally conformal almost cosymplectic manifolds

Example text

6. Exercises 43 the Taylor series for this function must have an infinite number of terms. Do you think your series would yield a reasonable approximation for / ( x ) at X = 0? Why or why not? 17. Expand y = R -\- Sx -\- Tx^ about x = a (symbolically), and show that the Taylor series expansion reduces to the original polynomial. 18. Given: The derivative of y = e^^ is be^^, and (as you may remember) for any function y and any constant c, d{cy) _ dx dy dx' Using these facts, expand y = ce^^ as a Maclaurin series (which means you set a = 0).

C. Calculate the approximate value of p(0) that would be obtained from the three terms of the Taylor series, and determine the relative error of that approximation. D. 8. If those errors are relatively small, that would suggest (but not prove) that the integral of the three-term series would approximate the integral of the true p{z) reasonably well over this range. If not, you might want to add more terms, but as the higher derivatives get messy, we'll omit that here. Hints: To simphfy developing the Taylor series, you may find it helpful to give the leading constant in the formula for p{z) a symbolic name (like c); use a substitution like giz) = - z ^ / 2 , and make use of the chain rule.

To get dT/dV, we use the chain rule, here dT/dV = (dT/dC) x (dC/dV). Because T = f{C) and C = g{V), we can also write dT/dV = (df/dC) x {dg/dV)—this is just another way of saying the same thing. Often we need to combine rules. For example, let's differentiate yix) = 1 +£3x with a series of elemental steps to illustrate the process. Experienced mathematicians might perform many of these steps "in their heads," but here I illustrate the process in a way that even a novice can use safely: Let u 'd x^, from which du/dx = 3x^ Let f ^ 1 + e^^, s '4 ly and t il e^^ so v = s + t.