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Unformatted text preview: Mathematics 33B: Milestones for Week 2 Note: We start the Monday lecture off by doing the escape velocity prob- lem. Topic I. (Lectures 5 & 6): 2.6 exact differential equations. A. Total derivatives: g = g ( x,y ) is a function of two variables. The total derivative of g is dg = ∂g ∂x dx + ∂g ∂y dy . 1. g ( x,y ) = 2 xy 3 + e 4 x sin y . Compute dg . B. Differential Forms. We consider differential equations of the form P ( x,y )+ Q ( x,y ) d y d x = 0, where P and Q are functions of both the independent variable x and the dependent variable y . This is equivalent to the form P ( x,y ) dx + Q ( x,y ) dy = 0. C. We can use total derivatives to solve differential equations. 1. Recall some 32B and what it means for a differential form to be exact. A form ω = M ( x,y ) dx + N ( x,y ) dy is exact in a region R , if ω = df where f is some continuously differentiable function in R . Suppose M , N , ∂M ∂y and ∂N ∂x are all continuous in a disk D . The differential form ω is exact in D if and only if ∂M ∂y = ∂N ∂x . 2. Suppose that (1- sin x tan x ) dx + (cos x sec 2 y ) dy = 0. This is equivalent to the differential equation: 1- sin x tan x + cos x sec 2 2 y d y d x = 0. This differential form is exact. Thus, there is a function g ( x,y ) such that dg = (1- sin x tan x ) dx +(cos x sec 2 y ) dy . Then the condition that dg = 0 gives us by integration that g = C where C is a constant. g ( x,y ) = x +cos x tan y + K ....
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