for and . Also, assume and the partial derivative are both continuous functions on this rectangle. Suppose are differentiable real valued functions defined on , with values in (i.e. for every ). Now, set and Then, by properties of Definite Integrals, we can write Since the functions are all differentiable (see the remark at the end of the proof), by the Multivariable Chain Rule, it follows that is differentiable, and its derivative is given by the formula: Now, note that for every , and for every , we have that , because when taking the partial derivative with respect to of , we are keeping fixed in the expression ; thus the basic form of Leibniz's Integral Rule with constant limits of integration applies. Next, by the First Fundamental Theorem of Calculus, we have that ; because when taking the partial derivative with respect to of , the first variable is fixed, so the fundamental theorem can indeed be applied. Substituting these results into the equation for above gives: as desired. There is a technical point in the proof above which is worth noting: applying the Chain Rule to requires that already be differentiable. This is where we use our assumptions about . As mentioned above, the partial derivatives of are given by the formulas and . Since is continuous, its integral is also a continuous function, and since is also continuous, these two results show that both the partial derivatives of are continuous. Since continuity of partial derivatives implies differentiability of the function, is indeed differentiable. Three-dimensional, time-dependent form At time t the surface Σ in Figure 1 contains a set of points arranged about a centroid . The function can be written as with independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at . For a rigidly translating surface, the limits of integration are then independent of time, so: where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the integration to act on the integrand only: with the velocity of motion of the surface defined by This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference.