'Exact equation' presentation slideshows

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Some Achievements and Challenges of Integrability in AdS/CFT Correspondence

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2-4 Exact Equations

2-4 Exact Equations. 2-4-1 ?????. ?? first order DE ?????. ???. (1) ?. ????. ?????? Exact Equation ?????. (2) ?. is independent of x. ?. is independent of y. ??? Modified Exact Equation Method ?? ( ??? 2-4-5). 2-4-2 ?????. Review the concept of partial differentiation

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2-4 Exact Equations

2-4 Exact Equations. 2-4-1 ?????. ?? first order DE ?????. ???. (1) ?. ????. ?????? Exact Equation ?????. (2) ?. is independent of x. ?. is independent of y. ??? Modified Exact Equation Method ?? ( ??? 2-4-5). 2-4-2 ?????. Review the concept of partial differentiation

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3.1 Quadratic Functions

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EXAMPLE 1

EXAMPLE 1. Find the first order partial derivatives of (Chain rule) If then it may be possible to compute What can we say about an implicit function and the ODE:

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Ch 2.6: Exact Equations & Integrating Factors

Ch 2.6: Exact Equations & Integrating Factors. Consider a first order ODE of the form Suppose there is a function ? such that and such that ? ( x,y ) = c defines y = ? ( x ) implicitly. Then and hence the original ODE becomes Thus ? ( x,y ) = c defines a solution implicitly.

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Ch 2.6: Exact Equations & Integrating Factors

Ch 2.6: Exact Equations & Integrating Factors. Consider a first order ODE of the form Suppose there is a function ? such that and such that ? ( x,y ) = c defines y = ? ( x ) implicitly. Then and hence the original ODE becomes Thus ? ( x,y ) = c defines a solution implicitly.

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Ch 2.6: Exact Equations & Integrating Factors

Ch 2.6: Exact Equations & Integrating Factors. Consider a first order ODE of the form Suppose there is a function ? such that and such that ? ( x,y ) = c defines y = ? ( x ) implicitly. Then and hence the original ODE becomes Thus ? ( x,y ) = c defines a solution implicitly.

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Ch 2.6: Exact Equations & Integrating Factors

Ch 2.6: Exact Equations & Integrating Factors. Consider a first order ODE of the form Suppose there is a function ? such that and such that ? ( x,y ) = c defines y = ? ( x ) implicitly. Then and hence the original ODE becomes Thus ? ( x,y ) = c defines a solution implicitly.

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2.6 Rational Functions & Their Graphs

2.6 Rational Functions & Their Graphs. Objectives Find domain of rational functions. Use arrow notation. Identify vertical asymptotes. Identify horizontal asymptotes. Use transformations to graph rational functions. Graph rational functions. Identify slant (oblique) asymptotes.

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