Problem 1
Consider the motion of a thumbtack stuck in the tread of a tire which is on a bicycle moving at constant speed. This motion is given by the parametrized curve with .
(b). Find the Radius of Curvature of the Osculating Circle to at
(c). Find the Equation of the Osculating Circle to at
Problem 2
A particle moves so that its position vector is given by , where and is a constant.
(a). Find the Velocity and the Acceleration of the Particle
(b). Find the Speed of the Particle
(c). Find the Tangential Component of the Acceleration of the Particle
(d). Show that the Trajectory of This Particle Lies in a Plane
The torsion of the curve is zero at all points so it's trajectory must be in a plane.
Problem 3
Find the arc length of the curve for , where . Express your result in terms of , , and .
(Hint: The integral you get can be evaluated with a simple substitution. You may want to factor the integrand first.)
Problem 4
Find the value of , where and is the hemisphere
oriented so the surface normals point away from the centre of the hemisphere.
Define to be the circle of radius 2 centred at the origin in the -plane
Problem 5
Compute the net outward flux of the vector field
across the boundary of the region between the spheres of radius and radius centred at the origin.
Let be the sphere of radius 1 and be the sphere of radius 2 both centred at the origin.
Problem 6
Let .
(a). Find All Values of and for Which the Vector Field is Conservative
(b). If and Have the Values Found in (a), Find a Scalar Potential Function for
(c). Let Be the Curve with Parametrization from to . Evaluate
(Hint: The vector field in the above integral is not conservative, but it is almost equal to the conservative vector field ).
Problem 7
(a). Give Parametric Descriptions of the Form for the following Surfaces. Be Sure to State the Domains of Your Parameterizations
(i). The part of the Plane in the First Octant
(ii). The Cap of the Sphere for
(iii). The Hyperboloid for
(b). Use Your Parametrization from part (a) to Compute the Surface Area of the Cap of the Sphere for
Problem 8
Determine if each of the given statements is True or False. Provide a reason or a counterexample.
(a). A Constant Vector Field is Conservative on
True if then there exist a scalar function for which .
(b). If for All Points in the Domain of , then is a Constant Vector Field
No consider it's divergence is zero and it is non-constant.
(c). Let Be a Parametrization of a Curve in . if and Are Orthogonal at All Points of the Curve , then Lies on the Surface of a Sphere for Some
True
(d). The Curvature at a point on a Curve Depends on the Orientation of the Curve
False the orientation changes the sign but the curvature takes the absolute value removing the sign change.
(e). The Domain of a Conservative Vector Field Must Be Simply Connected
No it does not need to be take the gravitational field for example it is undefined at