# Vector Function

A vector function is a function of the form

with

# Continuity of Vector Function

The vector function is continuous at if are continuous at

Theorem is continuous at if and only if for each there exists such that for

Proof.

If

then If then

# Differentiability of a Vector Function

A vector function is differentiable at if are differentiable at In this case

This is so since

# Differentiation Formula

• for

• for

• for

• for

• for

Proof.

• Example If is differentiable and show that the scalar function is differentiable and

Proof. Note that Since the square root function is differentiable for it follows that the composite

is differentiable. Differentiating the identity we obtain

so we have

• Example Let and Then

Proof.

It follows from CAB minus BAC formula that this expression equals

# Integral

If is continuous on the interval we define

# Properties of Integral

• for

• for

• Proof. Set Then it follows from Cauchy-Schwarz inequality that

If the conclusion holds. If then

# Curve in

Let the position of the curve be given by

Its velocity vector is and its acceleration vector is Assume that The unit tangent vector is

Suppose that The principal normal vector is

Since it follows that

This shows that the principal normal vector is perpendicular to the unit tangent vector.

# Arc Length

The arc length of for is given by

It follows from the Fundamental Theorem of Calculus that is differentiable and

for all The scalar function is called the speed of Thus

It follows that for all i.e., is an increasing function. Therefore the inverse function of exists. (Thus is the unique such that ) The position vector may be regarded as a function of the arc length Consider the composition

Taking the derivative of with respect to

# Curvature

The curvature of a curve is the magnitude of the change in the tangent per unit length:

Since

it follows that

and so

Thus the curvature may be computed as function of

• Example Let be a positive constant. Find the curvature of the curve

We have

so

Consequently

# Normal

The tangent vector satisfies When this identity is differentiated with respect to we obtain

Therefore At a point where is nonzero, the principal unit normal is given by

Thus

# Acceleration

Since the velocity satisfies

the acceleration satisfies

In this way, the quantity is expressed as the tangential component of acceleration while the quantity is expressed as the normal component of acceleration. Since

the tangential component of acceleration is

The computation

and the fact show that

Therefore, the normal component of acceleration may be expressed as

# Curvature for a Plane Curve

Suppose that the plane curve is given by Regard it as a space curve its curvature equals

Suppose that the plane curve is the graph of the function It can be regarded as the parametric curve

Hence its curvature equals

Geometric Interpretation

the angle that the tangent makes with the x-axis

Since it follows that so

It follows from the chain rule that

Hence

The reciprocal

of the curvature is called the radius of curvature. A point at a distance from the curve in the direction of principal normal is called the center of curvature. The circle with center at center of curvature and radius is called the osculating circle (also called the circle of curvature.) Thus the position vector of the center of curvature is

# Center of Curvature of a Plane Curve

Show that the center of curvature of a plane curve is given by

# Binormal

Let be that tangent and the normal of a curve. The vector

is called the binormal. These vectors satisfy

Differentiating both sides of the relation with respect to we have:

Therefore

Differentiating both sides of the relation with respect to we have:

Therefore

It follows that the vector is perpendicular to both and we may therefore write

for some scalar function which is known as the torsion of the curve. We now show how this function may be computed as function of Differentiating both sides of

with respect to we obtain

Since

and since

it follows that

Also we have

Therefore

Consequently

# Formula of Frenet

Differentiating with respect to we have

Therefore can be expressed as

for some scalars To determine these scalars we have

and

Therefore

Summery:

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