Derive can do integral vector calculus in any orthogonal, curvilinear coordinate system.  For example, you can compute scalar or vector potentials in rectangular (Cartesian), polar-cylindrical, or spherical coordinates.


Use POTENTIAL to compute a scalar potential of a vector.  For example,

POTENTIAL([y^2·z^3, 2·x·y·z^3, 3·x·y^2·z^2])

simplifies to

    2  3 
 x·y ·z  


As with antiderivatives, scalar potentials are unique only to within a constant. Thus, POTENTIAL may give a different result than one derived manually.  Such additive constants can be well disguised -- particularly in expressions involving logarithms or inverse trigonometric functions.  However, the gradient of a potential should be equivalent to the original vector.


Not all vectors have an associated scalar potential.  POTENTIAL merely computes a certain line integral involving the given vector.  If there is no potential, then the gradient of the resulting scalar will not be equivalent to the original vector.  It is your responsibility to check this.  In the case of a vector of two or three elements, it may be easier to see if the CURL of the given vector is equivalent to zero, as is necessary and sufficient for the existence of a scalar potential.


POTENTIAL takes an optional second argument that is a vector specifying the starting coordinates for the line integrals.  This argument defaults to a vector of zeros.  An inappropriate choice may lead to an infinite or unknown potential.  Try other values if this happens.  What you want is starting coordinates at which the given gradient is simple and finite.  π/2 and other simple rational multiples of π are often good alternatives for angular starting coordinates.  Infinity is often a good starting nonangular coordinate when 0 is not.  1 is often a good starting coordinate if the given gradient involves logarithms of that coordinate.


POTENTIAL also takes an optional third argument that is a Cartesian coordinate vector or a coordinate geometry matrix (see Differential Vector CalculusDifferential_Vector_Calculus).  Three-dimensional rectangular coordinates is the default using successive variables x, y, and z.


Use VECTOR_POTENTIAL to compute a vector potential of a three-element vector.  For example,

VECTOR_POTENTIAL([x, 0, y - z])

simplifies to

⎡    2           ⎤
⎢   y            ⎥
⎢- ————, - x·z, 0⎥
⎣   2            ⎦


As with POTENTIAL, VECTOR_POTENTIAL takes optional second and third arguments for specifying the starting coordinates and a coordinate geometry matrix.


Vector potentials are unique only to within an arbitrary gradient.  Consequently, two equally valid vector potentials may be quite different.  However, the curls of both vector potentials should be equivalent to the original vector.


Not all vectors have an associated vector potential.  VECTOR_POTENTIAL merely computes a certain line integral involving the given vector.  If there is no vector potential, the curl of the resulting vector will not be equivalent to the original vector.  It is your responsibility to check this.  Alternatively, it may be easier to see if the DIV of the given vector is zero, as is necessary and sufficient for the existence of a vector potential.


Other Vectors and MatricesVectors_and_Matrices topics

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