hw3-difftop

# hw3-difftop - Math 343 Homework 3 Due Wednesday 1 To be...

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: Math 343 Homework 3 Due Wednesday February 17, 2010. 1 To be handed in for grading 1. Is the cuspidal cubic { ( x,y ) ∈ R 2 | y 2 = x 3 } a submanifold of R 2 ? Why or why not? Hint: Review the definition of a submanifold from Lecture 1. 2. Take a smooth real valued function f : R 2 → R , the graph of f graph( f ) := { ( x,y,z ) ∈ R 3 | z = f ( x,y ) } . The tangent plane to this surface at the point ( a,b,f ( a,b )) is given by the equation z = f ( a,b ) + ∂f ∂x ( a,b )( x- a ) + ∂f ∂y ( a,b )( y- b ) . The aim of this question is for you to recover this equation (first seen in multivariable calculus) using the more general setting of manifolds. (a) Let M = graph( f ). Carefully explain why M is a 2-dimensional manifold. (b) Given ( a,b,c ) ∈ M then T ( a,b,c ) M is defined to be T ( a,b,c ) M = im dφ ( a,b ) , where φ is a parametrization in a neighborhood of the point ( a,b ). From (a) explicitly compute dφ ( a,b ) and find a basis for im dφ ( a,b ) . (c) The best linear approximation of M at ( a,b,c ) is given by ( a,b,c )+ T ( a,b,c ) M . In our case, this can be rewritten as the equation of the plane at ( a,b,c ) parallel to T ( a,b,c ) M . Find the equation of this plane and show that it matches the equation given above. Hint 1: The equation of a plane at a point ( a,b,c ) ∈ R 3 perpendicular to the vector ~v = v 1 v 2 v 3 is given by v 1 ( x- a ) + v 2 ( y- b ) + v 3 ( z- c ) = 0. Hint 2: Review the cross product of two vectors and its geometric interpretation. 3. Find the equation of the tangent plane to x 2 + y 2- z 2 =- 1 at the point (2 , 2 ,- 3)....
View Full Document

## This note was uploaded on 03/06/2010 for the course DEPARTMENT math434 taught by Professor Elizabethdenne during the Spring '10 term at Smith.

### Page1 / 5

hw3-difftop - Math 343 Homework 3 Due Wednesday 1 To be...

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

View Full Document
Ask a homework question - tutors are online