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The scalar product of two vectors can be constructed by taking the component of one vector in the direction of the other and multiplying it times the magnitude of the other vector. This can be expressed in the form:
SKALAR PRODUCT
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If the vectors are expressed in terms of unit vectors i, j, and k along the x, y, and z directions, the scalar product can also be expressed in the form:
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The vectors A and B cannot be unambiguously calculated from the scalar product and the angle. If the angle is changed, then B will be placed along the x-axis and A in the xy plane.
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Geometrically, the scalar product is useful for finding the direction between arbitrary vectors in space. Since the two expressions for the product:
Involve the components of the two vectors and since the magnitudes A and B can be calculated from the components using:
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The collection of partial derivative operators
the direction is given by the right-hand rule. If the vectors are expressed in terms of unit vectors i, j, and k in the x, y, and z directions, can be expressed in form:
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The vector product is compactly stated in the form of a determinant which for the 3x3 case has a convenient evaluation procedure
Once the scheme for determinant evaluation is familiar, this is a convenient way to reconstruct the expanded form:
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In cylindrical polar coordinates:
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The divergence of a vector field
in rectangular coordinates is defined as the scalar product of the del operator and the function
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In cylindrical polar coordinates:
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The curl of a vector function is the vector product of the del operator with a vector function:
The Curl
where i,j,k are unit vectors in the x, y, z directions. It can also be expressed indeterminant form
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The curl in cylindrical polar coordinates, expressed in determinant form is
The curl in spherical polar coordinates, expressed in determinant form is
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The LaPlacian
The divergence of the gradient of a scalar function is called the Laplacian. In rectangular coordinates:
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In cylindrical polar coordinates:
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CARTESIAN COORDINATE SYSTEM
 In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.
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In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.
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The most common coordinate system for representing positions in space is one based on three perpendicular spatial axes generally designated x, y, and z
Any point P may be represented by three signed numbers, usually written (x, y, z) where the coordinate is the perpendicular distance from the plane formed by the other two axes.
Often positions are specified by a position vector r which can be expressed in terms of the coordinate values and associated unit vectors.
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Although the entire coordinate system can be rotated, the relationship between the axes is fixed in what is called a right-handed coordinate system.
The distance between any two points in rectangular coordinates can be found from the distance relationship
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is given by
POLAR COORDINATE SYSTEM
A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line).
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Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates (r, θ) there is a single point, but any point is represented by many pairs of coordinates. For example (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for the same point. The pole is represented by (0, θ) for any value of θ.
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With the axis of the circular cylinder taken as the z-axis, the perpendicular distance from the cylinder axis is designated by r and the azimuthal angle taken to be Φ
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Spherical coordinate systems
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Coordinate curves and surfaces
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Coordinate surfaces in the Spherical coordinate system