 If you understand the divergence integral theorem, then the curl integral theorem should not be totally cryptic to you. You already know the vector field F, the scalar product, the nabla operator and the dA element should also be familiar to you now. First, let's look at the right hand side of the equation. The L is a line in space. The circle around the integral sign indicates that this line must be closed. That is, it should form a loop whose beginning and end are connected. The dL is an infinitesimal line element of the loop, so an infinitely small piece of the line. Again, you should notice that the dL element is shown in boldface. It's a vector with a magnitude and a direction. The magnitude of the dL element indicates the length of this small line and its direction points along the line L. Now, the scalar product between the vector field F and the line element dL is formed. You already know what the task of the scalar product is. First, split up the vector field into two parts. Into F parallel, which is parallel to the dL element, and into F orthogonal, which is orthogonal to the dL element. The scalar product with a dL element eliminates the orthogonal component without touching the part of the vector field parallel to the dL element. Since at each location the dL element points along the line, only the part of the vector field that runs along the line L is considered in the scalar product. The other part of the vector field drops out. Then the scalar products for each location on the loop are summed up using the integral. Such an integral in which small line elements are summed up is called line integral. Now you know what happens on the right hand side of the curl integral theorem. The line integral measures how much of the vector field F runs along the line L. Because the line is closed, this scalar product returns to the same point where the summation started. The closed line integral thus indicates how much of the vector field F rotates along the loop L. If the vector field F in this line integral is an electric field E, then this line integral is referred to as electric voltage along the line L. On the other hand, when the vector field F is a magnetic field B, the line integral is called magnetic voltage along the line L. The voltage in the case of an electric field is proportional to the energy that a positively charged particle gains as it passes the line L. A negatively charged particle on the other hand loses this energy as it passes the line L. The line integral of the electric field, that is the voltage, measures the energy gain or energy loss of charged particles as they pass through the line L under consideration. Now you should have understood the right hand side of the curl integral theorem. Let's look at the left hand side now. Here the surface A occurs again. This surface, unlike the divergence integral theorem, must not be a closed surface, but it is simply the surface enclosed by the line L. The A is again an infinitesimal piece of the surface A, and at any location it is orthogonal on that surface. In addition, here comes the cross product which you have already met when we discussed the magnetic force. Here the cross product is formed between the nubla operator and the vector field F. In addition to the scalar product, it is the second way to multiply two vectors. This cross product between the nubla operator and the vector field F is called the curl of the vector field F. The result, in contrast to the scalar product, is again a vector field. This new vector states how much of the field F rotates around a point within the surface A. Then the scalar product is formed between the new vector field nubla cross F with the infinitesimal surface element DA. Thus, as you already know, only the part of nubla cross F is picked out which runs parallel to the surface element. Since the surface element is orthogonal to the surface A, the scalar product picks out only the part of the vector field nubla cross F, which is also orthogonal on the surface A. Subsequently, all scalar products within the surface A are summed up by means of the integral. Let's summarize the statement of the curl integral theorem. On the right-hand side, the vector field F is summed up along a line L. Thus, the rotation of the vector field around the enclosed surface is considered. On the left-hand side, the curl of the vector field F is summed up at each individual point within the surface. Both sides should be equal according to the theorem. The curl integral theorem thus states that the total curl of a vector field F within the surface A corresponds to the rotation of the vector field F along the edge L of that surface. Well, it is somehow clear that the rotation of the vector field inside of the surface cancels in the summation and only the rotation of the vector field along the edge L remains.