Lebesgue differentiation theorem for BV functions of several variables, and applications

We generalize the classical Lebesgue differentiation theorem for BV functions of one variable to BV functions of several variables. More precisely, using the notion of joint derivative and joint monotonicity, and through a Jordan decomposition, we show that a BV function of several variables has well-defined joint derivative almost everywhere. As a consequence, we derive the Newton-Leibniz formula for absolutely continuous functions of several variables.