 This is what the divergence integral theorem looks like in its full splendor. First, let's look at the right-hand side of the equation. The A represents a surface in closing any volume. For example, the surface of a cube, a sphere or the surface of any three-dimensional body you can think of. The small circle around the integral indicates that this surface must satisfy a condition. The surface must be closed. In other words, it must not contain any holes, so that the equality is met mathematically. The surface A is thus a closed surface. The F is a vector field and represents either the electric field or the magnetic field when considering the Maxwell equations. So it is a vector with three components. DA is an infinitesimal surface element that is an infinitely small surface element of the considered surface A. As you may have already noticed, the A in the DA element is shown in boldface. So it is a vector with a magnitude and a direction. The magnitude of the DA element indicates the area of the small piece of the surface. The DA element is orthogonal to the surface and by definition points out of the surface. The dot between the vector field and the DA element represents the so-called scalar product. The scalar product is a way to multiply two vectors. So here, the scalar product between the vector field and the DA element is formed. The scalar product is defined as follows. As you can see from the definition, the first, second and third components of the two vectors are multiplied and then added up. The result of the scalar product is no longer a vector but an ordinary number, a so-called scalar. To understand what this number means, you must first know that any vector can be written as the sum of two other vectors. One vector that is parallel to the DA element, let's call it F parallel. And another vector that is orthogonal to the DA element, let's call it F orthogonal. Another mathematical fact is that the scalar product of two orthogonal vectors always yields zero, which means that in our case, scalar product between the part F orthogonal and the DA element is zero. However, the scalar product between the part F parallel and the DA element is generally not zero. So now you can see what the scalar product on the right-hand side of the equation does. It just picks out the part of the vector field that is exactly parallel to the DA element. The remaining part of the vector field that points in the orthogonal direction is eliminated by the scalar product. Subsequently, the scalar products for all locations of the considered surface A are added up. That is the task of the integral. The right-hand side of the divergence integral theorem thus sums up all the components of the vector field F that flow into or flow out of the surface A. Such an integral in which small pieces of a surface are summed up is called surface integral. If, as in this case, the integrand is a vector field, this surface integral is called the flux phi of the vector field F through the surface A. This description is based on what this surface integral means. It measures how much of the vector field F flows out or flows into a considered surface A. If the vector field F in this surface integral is an electric field E, then this surface integral is called electric flux through the surface A. And if the vector field F is a magnetic field B, the surface integral is called magnetic flux through the surface A. Now let's look at the left-hand side of the theorem. V is a volume, but not any volume. It is the volume enclosed by the surface A. dV is an infinitesimal volume element, in other words, an infinitely small volume piece of the considered volume V. The upside-down triangle is called nabla operator, and it has three components like a vector. Its components, however, are not numbers, but derivatives corresponding to the space coordinates. The first component is the derivative with respect to x, the second component is the derivative with respect to y, and the third component is the derivative with respect to z. Such an operator, like the nabla operator, only takes effect when applied to a field, and that also happens in this integral. The nabla operator is applied to the vector field by taking the scalar product between the nabla operator and the vector field. As you can see, it is the sum of the derivatives of the vector field with respect to the space coordinates x, y, and z. Such a scalar product between the nabla operator and a vector field F is called the divergence of the vector field F. The result at the location x, y, z is no longer a vector, but a scalar which can be either positive, negative, or zero. If the divergence at location x, y, z is positive, then there is a source of a vector field F at this location. If this location is enclosed by a surface, then the flux through the surface is also positive. The vector field, so to speak, flows out of the surface. If the divergence at location x, y, z is negative, then there is a sink of a vector field F at this location. If this location is enclosed by a surface, then the flux through the surface is also negative. The vector field flows into the surface. If the divergence at location x, y, z disappears, then that location is neither a sink nor a source of the vector field. The vector field does not flow out or into, or it flows in as much as out, so the two amounts cancel each other out. Subsequently, the divergence, that is, the sources and sinks of the vector field, is summed up at each location within the volume using the integral. Such an integral where small pieces of volume are summed up is called volume integral. So let's summarize the statement of the divergence integral theorem. On the left side is the sum of the sources and sinks of the vector field within a volume, and on the right side is the total flux of the vector field through the surface of that volume, and the two sides should be the same. The divergence integral theorem thus states that the sum of the sources and sinks of a vector field within a volume is the same as the flux of the vector field through the surface of that volume.