to the equation where is an integer. All the points where are solutions to the equation where is an integer. Calculus Calculus can be applied to equations expressed in polar coordinates. The angular coordinate φ is expressed in radians throughout this section, which is the conventional choice when doing calculus. Differential calculus Using and , one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, u(x,y), it follows that (by computing its total derivatives) or Hence, we have the following formulae: Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,φ), it follows that or Hence, we have the following formulae: To find the Cartesian slope of the tangent line to a polar curve r(φ) at any given point, the curve is first expressed as a system of parametric equations. Differentiating both equations with respect to φ yields Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point : For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see curvilinear coordinates. Integral calculus (arc length) The arc length (length of a line segment) defined by a polar function is found by the integration over the curve r(φ). Let L denote this length along the curve starting from points A through to point B, where these points correspond to φ = a and φ = b such that . The length of L is given by the following integral Integral calculus (area) Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where . Then, the area of R is This result can be found as follows. First, the interval is divided into n subintervals, where n is some positive integer. Thus Δφ, the angle measure of each subinterval, is equal to (the total angle measure of the interval), divided by n, the number of subintervals. For each subinterval i = 1, 2, ..., n, let φi be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(φi), central angle Δφ and arc length r(φi)Δφ. The area of each constructed sector is therefore equal to Hence, the total area of all of the sectors is As the number of subintervals n is increased, the approximation