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Curl of a Vector Field
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Divergence of a Vector Field
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Gradient of a Scalar Field
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Properties of Transposes
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The Transpose of a Matrix
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Cartesian Coordinate
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Cylindrical Coordinate
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Spherical Coordinate
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Transform from Cartesian to Cylindrical Coordinate
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Transform from Cartesian to Spherical Coordinate
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Transform from Cylindrical to Cartesian Coordinate
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Transform from Spherical to Cartesian Coordinate
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Divergence Theorem/Gauss' Theorem
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Stokes' Theorem
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Definition of a Matrix
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Linear Algebra
> Cartesian Coordinate
Cartesian Coordinate
The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
A coordinate system consists of four basic elements:
Choice of origin
Choice of axes
Choice of positive direction for each ax
Choice of unit vectors for each axes
Infinitesimal Line Element:
Consider a small infinitesimal displacement
d
s
between two points
P
_{1}
_{ }
and
P
_{2}
(Figure1). In Cartesian coordinates this vector can be decomposed into:
Figure 1:
Displacement between two points.
Infinitesimal Area Element:
An infinitesimal area element of the surface of a small cube is given by
dA= (dx) (dy)
Area elements are actually vectors where the direction of the vector
is perpendicular to the plane defined by the area. Since there is a choice of direction, we shall choose the area vector to always point outwards from a closed surface, defined by the righthand rule. So for the above, the infinitesimal area vector is:
Infinitesimal Volume Element:
An infinitesimal volume element in Cartesian coordinates is given by
dV = dx dy dz
The position of any point on the Cartesian plane is described by using two numbers: (
x
,
y
). The first number
x
is the horizontal position of the point from the origin. It is called the
x
coordinate
. The second number,
y
, is the vertical position of the point from the origin. It is called the
y
coordinate
. Since a specific order is used to represent the coordinates, they are called
ordered pairs
For example, the ordered pair (5, 9) represents point 5 units to the right of the origin in the direction of the
x
axis and 9 units above the origin in the direction of the
y
axis.
In a system formed by a point,
O
, and an orthonormal basis at each point,
P
, there is a corresponding vector
in the plane such that:
The coefficients x and y of the linear combination are called coordinates of point P.
The first x is the abscissa and the second y is the ordinate. As the linear combination is unique, each point corresponds to a pair of numbers and each pair of numbers to a point.
Using Cartesian coordinates on the plane, the distance between two points (
x
_{1}
,
y
_{1}
) and (
x
_{2}
,
y
_{2}
) is defined by the formula
,
which can be viewed as a version of the Pythagorean Theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula
θ = tan1(m),
where
m
is the slope of the line.
Midpoint formula
If (x
_{1}
,y
_{1}
) and (x
_{2}
,y
_{2}
) are two points, the midpoint of the line joining these two points is given by:
Example 1:
Plot the graph of Cartesian coordinate points A (2, − 4), B (4, 3), C (−2, 3) and D (−3, −4).
Solution:
As shown in the figure:
A (2, − 4) lies in the fourth quadrant,
B (4, 3) lies in the first quadrant,
C (−2, 3) lies in the second quadrant,
D (−3, −4) lies in the third quadrant.
Example 2:
Find the Cartesian coordinate distance between the points A (3, 6) and B (6, 3).
Solution
:
Let "d" be the distance between A and B.
(x
_{1}
,y
_{1}
) = (3, 6) and (x
_{2}
,y
_{2}
) = (6,3)
Then d (A, B) is determined by:
Example 3:
Find the equation of the line having slope 7 and yintercept 3
Solution
:
Applying the slopeintercept formula, the equation of the line is given by:
y = mx + c
, where m = 7, and c = 3.
Insert the values we have:
y = 7x + (3)
y = 7x  3
Rearranging the above equation, we get
7x  y  3 = 0.
Example 4:
Determine the midpoint coordinates of a line segment joining points A (6, 8) and B (1,5)
Solution
:
We know that:
x
_{1}
= 6
x
_{2}
= 1
y
_{1}
= 8
y
_{2}
= 5
The required mid point is given by:
Let the midpoint be denoted by M (a, b), we have:
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