Symbolic Operations in Maxima

You can prove formulas involving vectors and differentiation. Let us begin with loading the package necessary to work with vectors.

--> load(vect)

We can define vectors as follows.

--> uu: [u1,u2,u3]
--> vv: [v1,v2,v3]
--> ww: [w1,w2,w3]

When you want to establish the equation

$\displaystyle \mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}) = (\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}$

We can calculate the left- and righ-hand side of the formula symbolically.

--> express(uu.(vv~ww))
--> express((uu~vv).ww)

It is often easy to compare the output, and establish the equality. The next example is to show that

$\displaystyle \mathbf{u}\times(\mathbf{v}\times\mathbf{w}) = (\mathbf{u}\cdot\mathbf{w})\mathbf{v} - (\mathbf{u}\cdot\mathbf{v})\mathbf{w}$

it is not clear whether the two sides of the formula are equal or not due to the different ways of expansion. Then we need to expand them completely and take the difference.

--> F1: express(uu~(vv~ww))
--> F2: (uu.ww)*vv - (uu.vv)*ww
--> expand(F1) - expand(F2)

Derivative formulas can be shown symbolically. First we have to declare that the symbols f and g are functions of t.

--> depends(f,t)
--> depends(g,t)

We can differentiate them with respect to t

--> diff(f*g, t)
--> diff(f/g, t)

Similarly we can work with vector functions. Again first we have to declare them as functions of t

--> depends(uu,t)
--> depends(vv,t)

Then we can calculate the left and the right side of the formula

$\displaystyle \frac{d}{dt}[\mathbf{u}\cdot\mathbf{v}] = \mathbf{u}'\cdot\mathbf{v} + \mathbf{u}\cdot\mathbf{v}'$

and compare the output to establish the equality.

--> diff(uu.vv, t)
--> diff(uu,t).vv + uu.diff(vv,t)

The following wxm files include the above demonstrations of vector triple product formula and of differentiation rules symbolically.

exercise-hw01.wxm (Proof of vector triple product formula)
theorem13-2.wxm (Proof of differentiation rules of dot product and of cross product)
theorem13-3.wxm (Proof of the derivative formula for unit tangent vector)