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lecture05

# lecture05 - Section 2.4 Exact Equations Lecture 05 Brief...

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Section 2.4: Exact Equations Jan 29, 2008 Lecture 05

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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

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Example, cont’d (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, cont’d We prefer to write this equation in the form 2 xdx + 2 ydy = 0 Claim: In general starting with F ( x , y ) = C

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lecture05 - Section 2.4 Exact Equations Lecture 05 Brief...

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