2010-03-17 - MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI...

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MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1. Review: Linear Algebra IV – Eigenvalues and Eigenvectors 1 1.1. Definition 1 1.2. Computation of eigenvalues and eigenvectors 1 2. Functions of several variables I: Introduction 1 2.1. Introductory Example 1 2.2. Functions of two or more independent variables 3 2.3. The level set 7 2.4. Limits and continuity 9 1. Review: Linear Algebra IV – Eigenvalues and Eigenvectors 1.1. Definition. Definition: Let A is an n × n square matrix . A vector v that is non-zero vector of size n × 1 and a number λ satisfy the equation Av = λv . Then we call v is the eigenvector of A and λ is the eigenvalue of A . 1.2. Computation of eigenvalues and eigenvectors. Result: The number λ is an eigenvalue of the square matrix A if and only if it satisfies the equation det( A - λI ) = 0 . If λ is an eigenvalue of the square matrix A then we find the corresponding eigenvector(s) by solving the linear system of equations ( A - λI ) v = 0 . NOTE: Any scalar multiple of an eigenvector is again an eigenvector. 2. Functions of several variables I: Introduction 2.1. Introductory Example. To survive in cold temperatures, humans must maintain a suffi- ciently high metabolic rate, or regulate heat loss by covering their skin with insulating material. There is a functional relationship that gives the lowest temperature for survival ( T e ) as a function of metabolic heat production ( M ) and whole-body thermal conductance ( g Hb ). The metabolic heat production depends on the type of activity; some values for humans are summarized in the following table: Date : 2010-03-17. 1
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2 JING LI Activity M in Wm - 2 Sleeping 50 Working at a desk 95 Level walking at 2.5 mph 180 level walking at 3.5 mph with a 40-lb pack 350 Table 1. Data adapted from Landsberg(1996) The whole body thermal conductance g Hb describes how quickly heat is lost. The value of g Hb depends on the type of protection; for instance, g Hb = 0 . 45 molm - 2 s - 1 without clothing, g Hb = 0 . 14 molm - 2 s - 1 for a wool suit, g Hb = 0 . 04 molm - 2 s - 1 for warm sleeping bag. That is, the smaller
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2010-03-17 - MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI...

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