By Ron Larson, Bruce H. Edwards

**Read Online or Download Multivariable Calculus, Ninth Edition PDF**

**Best calculus books**

**Plane Waves and Spherical Means: Applied to Partial Differential Equations**

The writer wish to recognize his legal responsibility to all his (;Olleagues and acquaintances on the Institute of Mathematical Sciences of recent York collage for his or her stimulation and feedback that have contributed to the writing of this tract. the writer additionally needs to thank Aughtum S. Howard for permission to incorporate effects from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for his or her cooperation and aid, and especially Lipman Bers, who urged the booklet in its current shape.

**A Friendly Introduction to Analysis**

This ebook is designed to be an simply readable, intimidation-free advisor to complicated calculus. principles and techniques of evidence construct upon one another and are defined completely. this is often the 1st e-book to hide either unmarried and multivariable research in this kind of transparent, reader-friendly environment. bankruptcy issues conceal sequences, limits of features, continuity, differentiation, integration, countless sequence, sequences and sequence of services, vector calculus, features of 2 variables, and a number of integration.

This e-book, meant as a pragmatic operating advisor for calculus scholars, comprises 450 routines. it really is designed for undergraduate scholars in Engineering, arithmetic, Physics, or the other box the place rigorous calculus is required, and may significantly gain a person looking a problem-solving method of calculus.

- Calculus, Better Explained: A Guide To Developing Lasting Intuition
- Distributions, Sobolev spaces, elliptic equations
- Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations
- Fourier Transforms

**Additional info for Multivariable Calculus, Ninth Edition**

**Example text**

Show that if X ∼ ????(????, ???? 2 ) then for ???? ∈ { − 1, 1}, ????[max{????(e − K), 0}] = ????e X ????+ 21 ???? 2 ( Φ ????(???? + ???? 2 − log K) ???? ) ( − ????KΦ ????(???? − log K) ???? ) where K > 0 and Φ(⋅) denotes the cumulative standard normal distribution function. Solution: We first let ???? = 1, ∞ ????[max{eX − K, 0}] = ∫log K (ex − K) fX (x) dx ∞ = −1 1 e 2 (ex − K) √ ∫log K ???? 2???? ∞ = By setting ???? = ∫log K −1 1 e 2 √ ???? 2???? ( ) x−???? 2 +x ???? ( ) x−???? 2 ???? dx ∞ dx − K ∫log K −1 1 e 2 √ ???? 2???? ( ) x−???? 2 ???? x−???? and z = ???? − ???? we have ???? ∞ ????[max{eX − K, 0}] = ∫ log K−???? ???? ∞ 1 2 1 2 1 1 e− 2 ???? +????????+???? d???? − K e− 2 ???? d???? √ √ log K−???? ∫ 2???? 2???? ???? dx.

Minimum and Maximum of Two Correlated Normal Distributions. Let X and Y be jointly normally distributed with means ????x , ????y , variances ????x2 , ????y2 and correlation coefficient ????xy ∈ (−1, 1) such that the joint density function is fXY (x, y) = 2????????x ????y 1 √ − e 1 2(1−????2xy ) [ ( 1 − ????2xy x−????x ????x )2 ( −2????xy x−????x ????x ] )( y−???? ) ( y−???? )2 y y + ???? ???? y y . Show that the distribution of U = min{X, Y} is ????xy ????y ⎞ ⎛ ⎛ −u + ???? + ????xy ????x (u − ???? ) ⎞ x y ⎟ ????y ⎜ −u + ????y + ????x (u − ????x ) ⎟ ⎜ fU (u) = Φ ⎜ √ √ ⎟ fX (u) + Φ ⎜ ⎟ fY (u).

Similarly we can also show that ⎞ ⎛ ⎜ ???? − ????xy y ⎟ ????(????, ????, ????xy ) = f (y)Φ ⎜ √ ⎟ dy ∫−∞ Y ⎜ 1 − ????2xy ⎟ ⎠ ⎝ ???? 1 2 1 where fY (y) = √ e− 2 y and ????(????, ????, ????xy ) + ????(−????, ????, −????xy ) = Φ(????). 2???? ◽ 17. Bivariate Normal Distribution Property. Let X and Y be jointly normally distributed with means ????x , ????y , variances ????x2 , ????y2 and correlation coefficient ????xy ∈ (−1, 1) such that the joint density function is [ fXY (x, y) = 2????????x ????y 1 √ − e 1 2(1−????2xy ) ( x−????x ????x )2 ( −2????xy x−????x ????x ] )( y−???? ) ( y−???? )2 y y + ???? ???? y 1 − ????2xy y .