Note that this is a vector equation, i . e . it has three scalar equations.
2.
The original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements.
3.
According to Villamil energetics can be mathematically characterised by scalar equations, and rigid dynamics by vector equations.
4.
Since these are vector equations, they still have the same form for in " any " direction.
5.
However you're neglecting to consider that that is a vector equation, and it also tells you the direction of the force.
6.
To justify this claim, we transform our order scalar equation into an order one vector equation by the usual reduction to a first order system.
7.
It is a vector equation, so it is to be split in 3 equations for each of the X, Y, Z components, yielding:
8.
Oliver Heaviside, later reformulated Maxwell's original equations into the set of four vector equations that are generally known today as Maxwell's equations.
9.
Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude.
10.
In three spatial dimensions, this is a system of three coupled second order ordinary differential equations to solve, since there are three components in this vector equation.
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