This chapter introduces the Gauss-Ostrogradski-Green's formula, which is a fundamental result in classical mathematical analysis. The formula states that the surface integral of a vector function over a closed surface \( S \) is equal to the volume integral of the divergence of the vector function over the region \( D \) enclosed by the surface. Mathematically, it is expressed as: \[ \oint_S [P(x, y, z) \mathrm{d}y \mathrm{d}z + Q(x, y, z) \mathrm{d}z \mathrm{d}x + R(x, y, z)] \mathrm{d}x \mathrm{d}y = \int_D \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) \mathrm{d}x \mathrm{d}y \mathrm{d}z. \] The chapter then derives specific forms of this formula useful in the theory of elliptic equations. It introduces the gradient, divergence, and curl operators for scalar and vector functions. For a vector function \( \vec{V} = V_1 \vec{i} + V_2 \vec{j} + V_3 \vec{k} \), the divergence and curl are defined as: \[ \operatorname{div} \vec{V} = \frac{\partial V_1}{\partial x} + \frac{\partial V_2}{\partial y} + \frac{\partial V_3}{\partial z}, \] \[ \operatorname{curl} \vec{V} = \left( \frac{\partial V_3}{\partial y} - \frac{\partial V_2}{\partial z} \right) \vec{i} + \left( \frac{\partial V_1}{\partial z} - \frac{\partial V_3}{\partial x} \right) \vec{j} + \left( \frac{\partial V_2}{\partial x} - \frac{\partial V_1}{\partial y} \right) \vec{k}. \] The formula is restated in terms of these operators, and the outward normal \( \vec{n} \) of the surface \( S \) is defined. The chapter also explores the case where \( \vec{V} \) is a function of two scalar functions \( \varphi \) and \( \psi \), leading to the expression: \[ \oint_S \varphi \frac{d \psi}{d n} \mathrm{~d} \sigma = \int_D \operatorname{div}(\varphi \operatorname{grad} \psi) \mathrm{d} v. \] Finally, the chapter provides alternative notations for the gradientThis chapter introduces the Gauss-Ostrogradski-Green's formula, which is a fundamental result in classical mathematical analysis. The formula states that the surface integral of a vector function over a closed surface \( S \) is equal to the volume integral of the divergence of the vector function over the region \( D \) enclosed by the surface. Mathematically, it is expressed as: \[ \oint_S [P(x, y, z) \mathrm{d}y \mathrm{d}z + Q(x, y, z) \mathrm{d}z \mathrm{d}x + R(x, y, z)] \mathrm{d}x \mathrm{d}y = \int_D \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) \mathrm{d}x \mathrm{d}y \mathrm{d}z. \] The chapter then derives specific forms of this formula useful in the theory of elliptic equations. It introduces the gradient, divergence, and curl operators for scalar and vector functions. For a vector function \( \vec{V} = V_1 \vec{i} + V_2 \vec{j} + V_3 \vec{k} \), the divergence and curl are defined as: \[ \operatorname{div} \vec{V} = \frac{\partial V_1}{\partial x} + \frac{\partial V_2}{\partial y} + \frac{\partial V_3}{\partial z}, \] \[ \operatorname{curl} \vec{V} = \left( \frac{\partial V_3}{\partial y} - \frac{\partial V_2}{\partial z} \right) \vec{i} + \left( \frac{\partial V_1}{\partial z} - \frac{\partial V_3}{\partial x} \right) \vec{j} + \left( \frac{\partial V_2}{\partial x} - \frac{\partial V_1}{\partial y} \right) \vec{k}. \] The formula is restated in terms of these operators, and the outward normal \( \vec{n} \) of the surface \( S \) is defined. The chapter also explores the case where \( \vec{V} \) is a function of two scalar functions \( \varphi \) and \( \psi \), leading to the expression: \[ \oint_S \varphi \frac{d \psi}{d n} \mathrm{~d} \sigma = \int_D \operatorname{div}(\varphi \operatorname{grad} \psi) \mathrm{d} v. \] Finally, the chapter provides alternative notations for the gradient