Same caution needs to be used when implementing in the calculator equations containing the square of a vector field: what the equation is exactly indicating can indeed be not so clear from the mathematical formulation.
Let's assume for example that you want to calculate the the dynamic pressure as a scalar:
dynamic_pressure = 0.5 * Density * Velocity^2 (eqn #1)
where velocity is a vector [vx,vy,vz] and Density is a scalar.
If you simply use the expression Velocity^2 in the calculator, you are actually using a vector with components [vx*vx, vy*vy, vz*vz] and with magnitude = sqrt(vx^4+vy^4+vz^4), and therefore the expression above in equation #1 would become:
dynamic_pressure = 0.5*Density*sqrt(vx^4+vy^4+vz^4)
and return a vector field instead of a scalar. The correct way to calculate this quantity is to use the expression:
dynamic_pressure = 0.5*Density*DOT(Velocity,Velocity) (eqn #2)
which will return a scalar of magnitude 0.5*Density*(vx^2+vy^2+vz^2)
The difference between the vector magnitude in equation #1 and the scalar in equation #2 becomes more pronounced as velocity gets larger.
On a more general note, please keep in mind:
Scalars that are equivalent to each other
- RMS(velocity) * RMS(velocity)
- (Velocity[x]*Velocity[x] + Velocity[y]*Velocity[y]+Velocity[z]*Velocity[z])
Vectors that are equivalent to each other.
- Velocity * Velocity. They both give you a vector of magnitude sqrt(vx^4+vy^4+vz^4)