Milestones_2_Math_33B_F08

Milestones_2_Math_33B_F08 - Mathematics 33B: Milestones for...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This note was uploaded on 02/08/2010 for the course MATH 33B taught by Professor Staff during the Winter '07 term at UCLA.

Page1 / 4

Milestones_2_Math_33B_F08 - Mathematics 33B: Milestones for...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online