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Summary of Math 223
- 1.
- Scalar variables: These are variables which hold a single
value, such as x, y, or t.
- 2.
- Vector variables (Chapter 12): These are variables which store several
pieces of information and are denoted by (in three space) or (in
n dimensional space.)
- 3.
- Scalar fields (Chapter 11): These are also referred to as functions. An
object such as f(x,y,z) assigns a single scalar value to each point
(x,y,z) in the domain of f.
- 4.
- Vector fields (17.1): A vector field such as assigns a vector
to each point in the domain.
- 5.
- Parameterized curves (16.1): A curve is a one dimensional object and
can be described by a single parameter, such as t. To describe a curve,
simply specify x = x(t), y = y(t), z = z(t) together with a range of
values for t (such as .) Optionally, one can use the
position vector, .
- 6.
- Parameterized surfaces: see section 16.3
- 7.
- The del operator (13.4-5, 20.1, 20.3): (also called nabla) is the operator . It has no meaning by itself and must
be used to operate on other objects.
- 1.
- Partial derviatives (13.1-2, 13.5): denotes the rate of change of f in the x-direction, similar derivatives
exist for the y and z directions;
- 2.
- Higher partial derivatives (13.7): since the partial derivatives
are functions, they too have derivatives; if the function is smooth, then
mixed partials are equal (ie. fxy = fyx, fxzz = fzxz =
fzzx;)
- 3.
- Gradients (13.4-5): grad . At each point, grad f tells you which direction f is
increasing the most, as well as the rate of increase in that direction;
- 4.
- Directional derivative (13.4-5): the rate of change of a function f
in the direction of the unit vector is ;
- 5.
- Laplacian (??): the Laplacian of a scalar field is simply
.
- 1.
- Divergence (20.1): The divergence of a vector field measures the flux
density at a point. It is represented by div which is defined to be
- 2.
- Curl (20.3): The curl of a vector field is a vector that points in the
direction perpendicular to the plan of maximum circulation density and has
magnitude equal to this maximum circulation density. In other words, it
points in the direction in which
is the greatest.
- 1.
- Velocity (16.2): The velocity vector, , points tangent to the
path, and is simply .
- 2.
- Speed (16.2): This is the magnitude of the velocity vector, .
- 3.
- Acceleration (16.2): this measures the change in the velocity vector;
.
- 4.
- Distance along the curve (16.2): if the path never crosses itself and there
is no value of t for which then the distance
along the curve from t = a to t = b is .
- 1.
- The volume beneath the surface f(x,y) that lies
above the region R (15.1)
in the xy-plane is where dA is the area element in
the coordinate system. In Cartesian, dA = dx dy. In polar
(15.5), . This is usually expressed as an iterated integral (15.2), for
example:
- 2.
- Triple integrals (15.3): these are simply a further extension of double
integrals. For an interpretion, let W be a three dimensional volume
over which the mass density of an object, is defined. Then
where dV is the appropriate volume element. In Cartesian, dV = dx dy
dz. In cylindrical (15.6), . In spherical (15.6), .
- 3.
- Averages: the average value of f(x,y) over the region R is
If the function is defined over a three dimensional region W then the
average of f over this region is
- 4.
- Area of a region: the total area of R is simply .
- 5.
- Volume: the total volume of a region W is .
- 1.
- Line integrals (18.1-2): These measure the amount of work which a vector
field ``does'' along a curve. If the curve, C, is parameterized by
then
- 2.
- Flux integrals (19.1-2): These measure how much a vector field flows through
an oriented surface. If the surface, S, is given by z = f(x,y) and lies
above a region R (this is the shadow of S) in the xy-plane then
if the surface is oriented upwards (towards +z axis.)
A point P is said to be a critical point of the function f(x,y,z) if
either
- 1.
- (ie. fx(P) = fy(P) = fz(P) = 0), or
- 2.
- fails to exist at point P.
Critical points come in three types. Maxima, minima, and saddle points.
- 1.
- A point P is a local maximum of f if f(x,y,z) < f(P) for all
(x,y,z) near P.
- 2.
- A point P is a local minimum of f if f(x,y,z) > f(P) for all
(x,y,z) near P.
- 3.
- A point P is a saddle point of f if there exist points Q and
Q' near P such that f(P) > f(Q) and f(P) < f(Q').
The second derivative test can be applied to classify these points. Look
at the quantities D = fxx fyy - fxy2 and fxx at the
critical point P. If
- 1.
- D > 0, fxx < 0 then P is a local maximum,
- 2.
- D > 0, fxx > 0 then P is a local minimum,
- 3.
- D = 0 then P is a saddle point,
- 4.
- D < 0 then the second derivative test provides no information.
Here we seek to find the maximum or minimum of some function f(x,y,z)
subject to the constraint that g(x,y,z) = c. This will occur at
solutions of the Lagrange multiplier problem
In this case, if the point is the solution for
a given value of c then
In other words, it tells you the rate of change of the maximal value of f
as the constraint increases (``how much bang for your buck.'')
A vector field is conservative if any of the following conditions
are met:
- 1.
- is independent of path,
- 2.
- the circulation of around every closed path is zero,
- 3.
- for some scalar potential function f(x,y,z).
Further, the curl test (20.3,20.5)can be applied to certain vector fields
to determine
whether the vector field is conservative. If is a smooth vector
field in three-space so that
- 1.
- , and
- 2.
- the domain,D, of is such that any closed curve that lies
in D can be smoothly contracted to a point while always remaining inside
D,
then is a gradient field.
A vector field is said to be a curl field if for some vector potential . The divergence test
can help determine if is a curl field. If is a smooth
vector field in three-space and
- 1.
- , and
- 2.
- the domain, D, of is such that every closed surface in
D encloses only points that also lie in D,
then is a curl field.
If C is a piecewise smooth oriented path from P to Q and f(x,y,z)
is a function whose gradient is continuous on C then
If W is a solid region whose boundary S is piecewise smooth and
oriented outward and is a smooth vector field defined over all of
S and W then
If C is a simple closed curve surrounding a region R in the plane,
oriented so that R is always on the left as you traverse C and is smooth and defined over all of R and C
then
If S is a smooth oriented surface with a piecewise smooth boundary C
oriented (from the orientation of S) by the right-hand rule and is a smooth vector field defined over all of S and C then
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Copyright © 1998 by Kris H. Green
The Vector Calculus Website at
http://www.math.arizona.edu/~vector
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