This chapter discusses elliptic partial differential equations and introduces key formulas from vector calculus. The Gauss-Ostrogradski-Green's formula is presented as a fundamental theorem in classical analysis, relating surface integrals to volume integrals. It is shown that this formula can be expressed in terms of vector operators such as divergence and curl. The gradient of a scalar function is defined as a vector of its partial derivatives, while the divergence of a vector function is the sum of its partial derivatives. The curl of a vector function is a vector whose components are the differences of partial derivatives of the vector components. The chapter then considers a vector function composed of scalar functions and derives an expression for the flux of this vector function through a surface. It also introduces the concept of the normal derivative of a scalar function, which is the directional derivative of the function in the direction of the surface normal. Using these concepts, the chapter derives a formula that relates the integral of a product of two scalar functions over a surface to an integral over the volume enclosed by the surface. The chapter also introduces the nabla operator, which is used to express the gradient, divergence, and curl in a more compact form. It shows that the divergence of a scalar times a gradient is equal to the dot product of the gradient of the scalar and the gradient of the other function, plus the scalar times the Laplacian of the other function. These formulas are essential in the study of elliptic partial differential equations.This chapter discusses elliptic partial differential equations and introduces key formulas from vector calculus. The Gauss-Ostrogradski-Green's formula is presented as a fundamental theorem in classical analysis, relating surface integrals to volume integrals. It is shown that this formula can be expressed in terms of vector operators such as divergence and curl. The gradient of a scalar function is defined as a vector of its partial derivatives, while the divergence of a vector function is the sum of its partial derivatives. The curl of a vector function is a vector whose components are the differences of partial derivatives of the vector components. The chapter then considers a vector function composed of scalar functions and derives an expression for the flux of this vector function through a surface. It also introduces the concept of the normal derivative of a scalar function, which is the directional derivative of the function in the direction of the surface normal. Using these concepts, the chapter derives a formula that relates the integral of a product of two scalar functions over a surface to an integral over the volume enclosed by the surface. The chapter also introduces the nabla operator, which is used to express the gradient, divergence, and curl in a more compact form. It shows that the divergence of a scalar times a gradient is equal to the dot product of the gradient of the scalar and the gradient of the other function, plus the scalar times the Laplacian of the other function. These formulas are essential in the study of elliptic partial differential equations.