lecture33-09

# lecture33-09 - 18.02 Multivariable Calculus(Spring 2009...

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

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.

Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 33 Line Integrals, conservative vector fields, Curl and potential functions in 3D April 30 Reading Material: From Simmons : 21.5. From Lecture Notes : V8, V11, V12. Last time: Divergence Theorem Today: Line Integrals, conservative vector fields, Curl and potential functions in 3D. 2 Line Integrals in 3D In this section we recall how we write line integrals for a 3D vector fields, the theorem about conservative vector fields and the fundamental theorem of line integrals. All three of these facts are just a generalization of what we know already in 2D. In this lecture we assume that all vector fields are continuously differentiable. Line Integrals in 3D : Assume ~ F = M ˆ i + N ˆ j + P ˆ k and that a curve C is parametrized by the vector position ~ r ( t ) = x ( t ) ˆ i + y ( t ) ˆ j + z ( t ) ˆ k for a ≤ t ≤ b then Z C ˆ F · d~ r = Z C M dx + N dy + P dz = b Z a M ( x ( t ) , y ( t ) , z ( t )) x ( t ) dt + N ( x ( t ) , y ( t ) , z ( t )) y ( t ) dt + P ( x ( t ) , y ( t ) , z ( t )) z ( t ) dt. Exercise 1. Consider the vector field ~ F = y ˆ i- x ˆ j + z ˆ k and the curve C given by ~ r ( t ) = cos t ˆ i + sin t ˆ j + t ˆ k for ≤ t ≤ 4 π . Compute the work of ~ F along C ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

lecture33-09 - 18.02 Multivariable Calculus(Spring 2009...

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

View Full Document
Ask a homework question - tutors are online