1
Scalar-Valued Function (SVF)
1.1
Introduction
It is a function that takes a vector as input and converts it to a
single real nunber as an ouput. This means that the function takes
one or more real values in the domain and converts to a single real
number. In general a n-variable scalar valued function acts as a
map from Rn to the real number line R. That is,
f : Rn ! R:
For example,
f (x) = x2 2 R : is a function of one variable
f (x; y) = (2x + y) 2 R: is a function of two variables
f (x; y; z) = x2 + 2yz 2 2 R: is a function of three variables
Parametric Equations
De…nition. A parametric equation is one in which the x and
y coordinates of a curve are both written as functions of another
variable called a parameter; this is usually given the letter t or .
For instance, instead of the equation y = x2, which is in Cartesian
form, the same equation can be described as a pair of equations in
parametric form: x = t and y = t2. This conversion to parametric form is called parameterization, which provides great e¢ ciency
when di¤erentiating and integrating curves. Parametric equations
are used in locating objects in motion. In this case the latitude,
longitude and altitude of the object is parametrized as functions
of time t. The overall idea of using parametric equations is mainly
for convenience.
Why Parametric Equations
There are a lot of curves that cannot be written as single equations in terms of only x and y, so to deal with such curves we
introduce parametric equations. Instead of de…ning y in terms of
x (y = f (x)) or x in terms of y (x = h (y)); we de…ne both x and
y in terms of a third variable t called a parameter as follows:
x = x (t) and y = y (t) :
Each value of t de…nes a point in 2-D as
(x; y) = hx (t) ; y (t)i
or a point in 3-D as
(x; y; z) = hx (t) ; x (t) ; z (t)i
that we can plot. The collection of points that we get, by letting
t be all possible values of R is the graph of the parametric equations
and is called a parametric curve.
2
Vector-Valued Functions (VVF)
2.1
Introduction
In Advanced Calculus 1 (Semester one), we studied Scalar Valued Functions (SVF) of several variables. We plotted real number
points in 2-D and 3-D directly without any calculations. For example we plotted points such as (2; 3) ; (1; 2) ; (3; 4) etc. on the 2-D
plane and (1; 2; 4) ; ( 3; 2; 1) etc. in 3-D space.
In this second semester we will study vector-valued functions
titled (Advanced Calculus 2). The components of a vector-valued
function are called parametric equations. For such functions we
do not plot the points directly. We can only get them by …rst
evaluating them from their parametric equations for di¤erent values
of t. For example a point in 2-D is given by
(x; y) = hx (t) ; y (t)i :
Similarly a point in 3-D is given by
(x; y; z) = hx (t) ; x (t) ; z (t)i
For example if,
x (t) = t + 1; y (t) = t2
4 and z (t) = t2;
Then for t = 1 we have
x = x (1) = t + 1 = 1 + 1 = 2
y = y (1) = t2 4 = 1 4 = 3
z = z (1) = t2 = 1
(x; y; z) = (2; 3:1)
Thus we can now plot the point (2; 3; 1) in 3-D which corresponds
to the parameter t = 1: When series of points corresponding to
di¤erent values of t are plotted, we get the graph of the vectorvalued function in space.
De…nition
A vector valued function is a function whose domain is a set of
real numbers say t and the range is a set of vectors with componets
de…ned as x (t) ; y (t) ; and z (t) which are real numbers after evaluating these functions at t 2 R. In general a vector-valued function
acts a map from the real line R to the vector Rn. That is,
r : R ! Rn :
Vector valued functions are therefore, simply an extension of
scalar functions, where both the domain and the range are the
set of real numbers.
In two dimensions
r : R ! R2 ;
where
r(t) = x(t)i + y(t)j or hx(t);y(t)i
In three dimensions
r : R ! R3 ;
r(t) = x(t)i + y(t)j + z(t)k or hx(t);y(t); z (t)i :
Remark
1. The axis (real line) of the parameter t is distinct from the coordinate axes of the plane in which the curve lies.
2. For vector valued functions, the values of r traces out a curve
called the plane curve in 2-D and space curve in 3-D.
3. Vector valued functions map real numbers to vectors with real
numbers as components, whiles scalar valued fucntions map real
numbers in vector form to single real nunbers.
4. Domain of the vector valued functions r are real numbers while
its range are vectors made up of real numbers.
5. The collection of these vectors when plotted gives a graph of
the curve of r.
6. To …nd the orientation of a curve, we plot points for increasing
values of t.
7. The notation h ; ; i is called the angle vector.
Space Curves and Parametric equations
For di¤erent values of t, we get the corresponding (x; y; z) coordinates, de…ned by the functions x (t ) ; y (t) and z (t) : The set of
generated points (x; y; z) ; de…nes a curve called the space curve C.
The equations for x (t ) ; y (t) and z (t) are called the parametric
equations of the curve C.
Parabola
A parabola is the set of points whose distance from a certain point
(the focus) is equal to their distance from a certain line (Directrix).
Conic Section (also called a quadrative curve)
It is a curve obtained as a result of the intersection of the surface
of a cone with a plane. Examples are: circle, ellipse, parabola and
hyperbola