lecture05 - Section 2.4: Exact Equations Jan 29, 2008...

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Unformatted text preview: Section 2.4: Exact Equations Jan 29, 2008 Lecture 05 Brief Vector Calc. What is the meaning of something like F ( x , y ) = C . 1. This is an implicit equation for y in terms of x . 2. This is a family of curves parametrized by C . 3. I can graph it using winplot 4. The vector calculus people call these curves the level sets of the function z = F ( x , y ). Lecture 05 Example: Let F ( x , y ) = x 2 + y 2 . (a) Describe the set of curves F ( x , y ) = C parametrized by C . (I used winplot for both the curves and phase plane.) Lecture 05 Example, contd (b) Find F x and F y . Note: F x is the derivative of F with respect to x , keeping y fixed. It is called the partial derivative of F wrt to x . In our case F ( x , y ) = x 2 + y 2 so F x = 2 x Also F y = 2 y (c) What is the DFQ satisfied by these curves? In our case: Starting with x 2 + y 2 = C , and performing implicit differentiation, we are led to: 2 x + 2 y dy dx = 0 Lecture 05 Example 1, contd...
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This note was uploaded on 04/07/2009 for the course MATH taught by Professor Lotfi during the Spring '09 term at University of Arizona- Tucson.

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lecture05 - Section 2.4: Exact Equations Jan 29, 2008...

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