Editing Planimeter (section)

==Mathematical derivation==
The operation of a linear planimeter can be justified by applying [[Green's theorem]], though the design of the major varieties predates the theorem's proof. Apply it to the components of the [[vector field]] N, given by:

:<math>\!\,N(x,y)=(b-y,x),</math>

where ''b'' is the ''y''-coordinate of the elbow E.

This vector field is perpendicular to the measuring arm EM:

:<math>\overrightarrow{EM}\cdot N = xN_x+(y-b)N_y=0</math>

and has a constant size, equal to the length ''m'' of the measuring arm:

:<math>\!\,\|N\| =\sqrt{(b-y)^2+x^2}=m</math>

Then:
:<math>
\begin{align}
& \oint_C(N_x \, dx + N_y \, dy) = \iint_S\left(\frac{\partial N_y}{\partial x}-\frac{\partial N_x}{\partial y}\right) \, dx \, dy \\[8pt]
= {} & \iint_S\left(\frac{\partial x}{\partial x}-\frac{\partial (b-y)}{\partial y}\right) \, dx \, dy = \iint_S  \, dx \, dy = A,
\end{align}
</math>

because:
:<math>\frac{\partial}{\partial y}(y-b) = \frac{\partial}{\partial y}\sqrt{m^2-x^2} = 0,</math>

The left hand side of the above equation, which is equal to the area ''A'' enclosed by the contour, is proportional to the distance measured by the measuring wheel, with proportionality factor ''m'', the length of the measuring arm.

The justification for the above derivation lies in noting that the linear planimeter only records movement perpendicular to its measuring arm, or when 
:<math>N\cdot(dx,dy)=N_xdx+N_ydy</math> is non-zero. When this quantity is integrated over the closed curve C, [[Green's theorem]] and the area follow.

=== Polar coordinates ===
The connection with Green's theorem can be understood in terms of [[Polar coordinate system#Integral calculus (area)|integration in polar coordinates]]: in polar coordinates, area is computed by the integral <math display="inline"> \int_\theta \tfrac{1}{2} (r(\theta))^2\,d\theta,</math> where the form being integrated is ''quadratic'' in ''r,'' meaning that the rate at which area changes with respect to change in angle varies quadratically with the radius.

For a [[parametric equation]] in polar coordinates, where both ''r'' and ''θ'' vary as a function of time, this becomes
<math display="block">\int_t \tfrac{1}{2} (r(t))^2 \, d(\theta(t)) = \int_t \tfrac{1}{2} (r(t))^2\, \dot \theta(t)\,dt.</math>

For a polar planimeter the total rotation of the wheel is proportional to <math display="inline"> \int_t r(t)\, \dot \theta(t)\,dt,</math> as the rotation is proportional to the distance traveled, which at any point in time is proportional to radius and to change in angle, as in the circumference of a circle (<math display="inline"> \int r\,d\theta = 2\pi r</math>).

This last integrand <math display="inline"> r(t) \,\dot \theta(t)</math> can be recognized as the derivative of the earlier integrand <math display="inline"> \tfrac{1}{2} (r(t))^2 \dot \theta(t)</math> (with respect to ''r''), and shows that a polar planimeter computes the area integral in terms of the ''derivative'', which is reflected in Green's theorem, which equates a line integral of a function on a (1-dimensional) contour to the (2-dimensional) integral of the derivative.