Milestones_2_Math_33B_F08

# Milestones_2_Math_33B_F08 - Mathematics 33B Milestones for...

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

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 .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern