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This is a good question. It is hard to explain the concept of magnetic flux without using surface integral. However, I'll give a try. Suppose you put a mesh in flowing water. How do you compute the water flowing through the mesh per second? The mesh is composed of many small hole. If you calculate the water flowing through each hole per second and sum them up, you'll get the water flowing through the whole mesh per second. Since each hole is so small, it can be regarded as within a plane, therefore each hole has an area. Although the velocity of water may differ at different positions, since the hole is very small, the velocity of water flowing through a hole can be regarded as not changing within the area of the hole. Therefore, if the velocity of the water flowing through a hole is perpendicular to the hole, the water flowing through the hole per second is just V*A, where V is the velocity, A is the area. However, if the angle of the velocity and the normal vector of the hole (the normal vector of the hole is a direction which is perpendicular to the aperture of the hole and which is pointed outward. You know the mesh has two sides. Before you begin your calculation you should specify one side as interior side and the other as exterior side so that for each hole you can specify its normal vector in a uniform way) is theta, the water flowing through the hole per second is V*A*cos(theta). This quantity (how much water flowing through the hole per second) is called the velocity flux through the hole. Some examples: if the water flowing perpedicularlly out of the hole, the velocity flux is V*A; if the water flowing perpendicularlly into the hole, the velocity flux is -V*A; if the water flowing parallelly to the hole, the velocity flux is 0, since no water flowing in or flowing out. After the velocity flux through each hole is calculated, the velocity flux through the whole mesh is just a summation of the velocity flux through all its holes. Now we replace the velocity with magnetic intensity (usually denoted as B, and is a vector field) in the above descriptions, we'll get the definition of magnetic flux. And in most cases the mesh is fictitious. Instead of saying magnetic flux through a mesh, people say magnetic flux through a surface. However, you know you can divide the surface into meshes and calculate the magnetic flux through each mesh cell and sum them up. Of couse if you divide the surface in different ways you may get different results, but if you divide it thinner and thinner, you'll see such differences become smaller and smaller. If you can divide the mesh into infinitely small cells, difinitely you'll get one certain value, that's the real value of magnetic flux. By dividing the surface into infinitely small cells and summing the magnetic flux through each cell up, actually you have done a surface integral. As to the "definition" of magnetic flux based on magnetic field lines, it's only an intutive description and can not be used for accurate calculations.

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