MATH 317

Vector

Vectors are any quantity that can not be represented by a single number. Vectors are typically represented by an arrow pointing in the direction of the vector with the length of the arrow representing the magnitude of the vector.

Example Vector

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Vectors can be written using different notations and have a set of intrinsic properties, these vectors can also be transformed using different operations.

Vector-Valued Function

Vector-valued functions are functions of one or more variables which outputs a vector. Formally a vector-valued function is any function in which where and .

Vector Field

A vector field is an assignment of a vector to every point in a given space, most commonly two- or three-dimensional Euclidean space. Formally, a vector field on a domain is a function , where each input point corresponds to a vector in the same dimension.

Examples

Non-examples

Not defined at all points in space

Parameterized Curve

A parameterized curve is a mathematical representation of a curve in which the coordinates of points on the curve are expressed as functions of one or more independent parameters, most commonly a single real parameter, usually denoted by .

Finding Parameterizations of Common Curves

Lines

Let be the line passing through the point that is parallel to the vector . We can then parameterize as follows:

Line Segments

Let be the line from the points to . We can the parameterize as follows:

Circles

Given a circle of radius centred at the origin in the -plane. We can parameterize the circle as follows:

Ellipse

Given a ellipse with two points at and centred at the origin in the -plane. We can parameterize the ellipse as follows:

Examples

Example 1

Find the parameterization of the curve given by the intersection of the cylinder and the plane .

The projection of onto the -plane is the unit circle so we can take

Then since we get

Example 2

Find a parameterization of the curve given by the intersection of the surfaces and .

TODO

Arc Length

The arc length is the distance between two points along a curve. Given a parameterized curve it's arc length from the point to is defined as follows:

Relation to Speed

In physical quantities if we consider the curve to be the position vector of a particle, it's derivative is therefore it's velocity, taking the size of this vector gives us the speed of the particle, the arc length is therefore just integrating the speed of a particle over time.

Examples

Find the arc length of one revolution of the helix for , where and are constants.

Parameterization by Arc Length

Parameterization by arc length is a method to parameterize a curve such that the resulting function has the parameter such that is the arc length of the original curve.

To parameterize a curve by arc length first the curve needs to be parameterized normally into some function , then using the equation for arc length we solve for giving us some function which gives us a function which is our curve parameterized by arc length.

Example

Re-parameterize the function in terms of arc length measured from .

Unit Tangent Vector

The unit tangent vector is the vector tangent to a curve scaled to a unit vector. Given a parameterized curve defined by we can express it's unit tangent vector as:

Alternatively because a curve that is parameterized by arc length is traced at unit speed we can express it as

Where must be some curve which is parameterized by arc length.

Principal Normal Vector

The principal normal vector is the direction the unit tangent vector is turning at a given point. Given a parameterized curve defined by we can express it's principal normal vector as:

Osculating Plane

The osculating plane of a curve is the linear span of the tangent and normal vectors of the curve.

Osculating Circle

The osculating circle of a curve at a given point is the circle that best approximates the curvature of the curve at that specific point. The osculating circle sits in the osculating plane of the curve at the same point.

The osculating circle at a point has a radius of and is centred at in the osculating plane.

Conservative Vector Field

A conservative vector field is any vector field that is the gradient of some function. The function that the field is the gradient of is called a potential function. Formally is a conservative vector field if there is some function such that and it's curl at every point must be zero.

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Line Integral

Line integrals are the natural generalization of the integral into the sum of some function over any arbitrary curve instead of just over an interval on the real line.

Line Integral of a Scalar Function

Formally the line integral of scalar function over the curve is defined as

Where is some parameterization of on the interval .

Derivation

We can use the same intuition that arises from approximating integrals with Riemann sum's to arise on our the definition of line integrals of scalar functions. Riemann sum's state that the sum of some function over a region can be written as an infinite sum of rectangles with the height of at some point in each infinitely small subinterval with width :

For a line integral, the key difference is that instead of summing over an interval on the real line, we sum over an arbitrary curve . Because of this difference we generalize to instead of being some small subinterval of the [Real Number|real] line to generally just represent some small segment of the curve we are summing over. We can represent this small segment of the curve as where is the arc length along the curve. This gives us the following definition of a line integral:

To evaluate this line integral we need to express in terms of the parameterization of the curve. We can do this by noting that the arc length along a curve is given by the integral of the magnitude of the derivative of the parameterization:

Differentiating both sides with respect to gives us:

This means that can be expressed as . Substituting this back into our definition of the line integral gives us:

Line Integral of a Vector Field

The line integral of a vector field over a curve is defined as

Where is some parameterization of on the interval .

Independence of Path

TO-WRITE

Derivation

Just as with the scalar case, we build intuition from a Riemann sum. This time however, the object we are summing is a vector field and we want to capture how much the field acts along the curve at each point, that is, we want to accumulate the component of in the direction of travel. This aim originates from the physical definitions of work.

At each point on the curve, the direction of travel is given by the unit tangent vector:

so the component of along the curve at each point is given by the dot product of with . Summing this scalar quantity over the curve with respect to arc length gives us:

Using the same substitution as in the scalar case, we can express in terms of the parameterization of the curve and we can express in terms of to get:

Surface Integral

Surface integrals are the natural generalization of the multiple integral into the sum of some function over any arbitrary surface instead of just over a region on a plane.

Surface Integral of a Scalar Function

Formally the surface integral of a scalar function over the surface is defined as

Where is some parameterization of over the region in the -plane, and and are the partial derivatives of with respect to and respectively.

Derivation

We can use similar intuition to line integrals to derive the definition of surface integrals. Just as line integrals generalize Riemann sums from intervals to curves, surface integrals generalize double integrals from planar regions to curved surfaces.

For a double integral over a region in the plane, we have:

Where represents small rectangular patches in the plane. For a surface integral, instead of summing over planar patches, we sum over small patches on the curved surface . We represent these surface patches as :

Where is a point on each surface patch. To evaluate this surface integral, we need to express in terms of a parameterization of the surface. Let be a parameterization of over a region in the -plane.

Consider a small rectangular patch in the -plane with dimensions by . This patch maps to a small curved patch on the surface . The edges of this rectangular patch in the parameter space map to curves on the surface. The displacement along the -direction (holding constant) is approximately:

Similarly, the displacement along the -direction (holding constant) is approximately:

These two displacement vectors span a parallelogram on the surface, and the area of this parallelogram approximates the surface area of the curved patch. The area of a parallelogram spanned by two vectors is the magnitude of their cross product:

Where is the area of the rectangular patch in the parameter space. This gives us the relationship between the surface area element and the parameter space area element:

Substituting this back into our definition of the surface integral gives us:

Surface Integral of a Vector Field

The surface integral of a vector field over a surface is defined as

Where is some parameterization of over the region in the -plane. Note that is a vector representing both the magnitude and orientation of the surface element.

Derivation

To derive the surface integral of a vector field, we use a similar approach to the surface integral of a scalar function. We start by approximating the surface integral using a sum over surface patches:

Where is a small vectorial surface element. Unlike the scalar case where we summed over scalar surface areas , here we sum over vector surface elements that capture both the area and the normal direction of each surface patch.

From our earlier derivation, we found that a small patch in the parameter space with dimensions by maps to a parallelogram on the surface spanned by and .

The vector area of this parallelogram is given by the cross product of these displacement vectors:

This gives us the vector surface element in differential form:

Note that is a vector perpendicular to the surface (a normal vector) with magnitude equal to the area scaling factor from the scalar case.

Substituting this back into our definition of the surface integral gives us:

Examples

Example 1

Find the flux across the piece of the cone , oriented with upward pointing unit normal.

Example 2

The temperature in a metal ball is proportional to the square distance of the distance from the centre of the ball. Find the rate of heat flow across a sphere of radius with centred at the centre of the ball.

Green's Theorem

Green's theorem relates line integrals over a closed curve to double integrals. Green's theorem is a special case of Stoke's theorem.

Formally Green's theorem states that given a closed region bounded by a positively oriented curve on plane for any vector field defined on we have the following relationship:

The path must always be counter clockwise.

Divergence Theorem

The divergence theorem states that the triple integral of the divergence of a vector field over a region is equal to the surface integral of a vector field over the boundary of , called .

For the divergence theorem the solid region doesn't need to be connected.

Examples

Example 1

Verify the divergence theorem with over the cube .

Split the cube into 6 different faces

Starting with the top face with the normal vector

By symmetry we can see that the same is true for all the sides.

Example 2

Use the divergence theorem to calculate the surface integral where and is the surface of the tetrahedron with vertices , , and with outward orientation.

Example 3

Compute where where is the upper unit hemisphere with upward pointing normal.

Consider as the circle disk with radius 1 defined around the origin in the -plane with downward orientation and the volume defined as .