General examples of usage

This section details the solution of two basic problems in vector math/calculus using the sympy.vector package.

Quadrilateral problem

The Problem

OABC is any quadrilateral in 3D space. P is the midpoint of OA, Q is the midpoint of AB, R is the midpoint of BC and S is the midpoint of OC. Prove that PQ is parallel to SR

Solution

The solution to this problem demonstrates the usage of Point, and basic operations on Vector.

Define a coordinate system

>>> from sympy.vector import CoordSys3D
>>> Sys = CoordSys3D('Sys')

Define point O to be Sys’ origin. We can do this without loss of generality

>>> O = Sys.origin

Define point A with respect to O

>>> from sympy import symbols
>>> a1, a2, a3 = symbols('a1 a2 a3')
>>> A = O.locate_new('A', a1*Sys.i + a2*Sys.j + a3*Sys.k)

Similarly define points B and C

>>> b1, b2, b3 = symbols('b1 b2 b3')
>>> B = O.locate_new('B', b1*Sys.i + b2*Sys.j + b3*Sys.k)
>>> c1, c2, c3 = symbols('c1 c2 c3')
>>> C = O.locate_new('C', c1*Sys.i + c2*Sys.j + c3*Sys.k)

P is the midpoint of OA. Lets locate it with respect to O (you could also define it with respect to A).

>>> P = O.locate_new('P', A.position_wrt(O) + (O.position_wrt(A) / 2))

Similarly define points Q, R and S as per the problem definitions.

>>> Q = A.locate_new('Q', B.position_wrt(A) / 2)
>>> R = B.locate_new('R', C.position_wrt(B) / 2)
>>> S = O.locate_new('R', C.position_wrt(O) / 2)

Now compute the vectors in the directions specified by PQ and SR.

>>> PQ = Q.position_wrt(P)
>>> SR = R.position_wrt(S)

Compute cross product

>>> PQ.cross(SR)
0

Since the cross product is a zero vector, the two vectors have to be parallel, thus proving that PQ || SR.

Third product rule for Del operator

See

WikiDel

https://en.wikipedia.org/wiki/Del

The Problem

Prove the third rule - \(\nabla \cdot (f \vec v) = f (\nabla \cdot \vec v) + \vec v \cdot (\nabla f)\)

Solution

Start with a coordinate system

>>> from sympy.vector import CoordSys3D, Del
>>> delop = Del()
>>> C = CoordSys3D('C')

The scalar field \(f\) and the measure numbers of the vector field \(\vec v\) are all functions of the coordinate variables of the coordinate system in general. Hence, define SymPy functions that way.

>>> from sympy import symbols, Function
>>> v1, v2, v3, f = symbols('v1 v2 v3 f', cls=Function)

v1, v2 and v3 are the \(X\), \(Y\) and \(Z\) components of the vector field respectively.

Define the vector field as vfield and the scalar field as sfield.

>>> vfield = v1(C.x, C.y, C.z)*C.i + v2(C.x, C.y, C.z)*C.j + v3(C.x, C.y, C.z)*C.k
>>> ffield = f(C.x, C.y, C.z)

Construct the expression for the LHS of the equation using Del().

>>> lhs = (delop.dot(ffield * vfield)).doit()

Similarly, the RHS would be defined.

>>> rhs = ((vfield.dot(delop(ffield))) + (ffield * (delop.dot(vfield)))).doit()

Now, to prove the product rule, we would just need to equate the expanded and simplified versions of the lhs and the rhs, so that the SymPy expressions match.

>>> lhs.expand().simplify() == rhs.expand().doit().simplify()
True

Thus, the general form of the third product rule mentioned above can be proven using sympy.vector.