First, let's define divergence. For vector field f (x, y, z) = (p (x, y, z), q (x, y, z), r (x, y, z)), its divergence at point M(x0, y0, z0) is defined as:
diff = _ P/_ x+_ Q/_ y+_ R/_ z
Where _ stands for partial derivative. Divergence describes the outflow or inflow of a vector field at a certain point. If the divergence is positive, it means that the vector field has a net outflow at this point. If the divergence is negative, it means that the vector field has a net inflow at this point; If the divergence is zero, it means that there is no net outflow or inflow of the vector field at this point.
Next, let's define the curl of a vector field f (x, y, z) = (p (x, y, z), q (x, y, z), r (x, y, z)), and its curl at point M(x0, y0, z0) is defined as:
rotF=[_Q/_x-_P/_y，_R/_x-_Q/_z，_P/_y-_R/_x]
Where [] stands for outer product. Curvature describes the rotation degree of the vector field at a certain point, and describes the rotation direction and speed of the vector field at that point.
The knowledge of differential geometry on a surface is needed to calculate the divergence and curl of a vector field on a surface. The specific steps are as follows:
1. Determines the parametric form of the surface. A surface can be described by one or more parametric equations. For example, the parameterized form of a sphere is (x-a) 2+(y-b) 2+(z-c) 2 = r 2, where (a, b, c) is the center coordinate of the sphere and r is the radius of the sphere.
2. Substitute the expression of vector field into parameterized form. Substituting the expression of vector field f into the parameterized form of the surface, the expression of f on the surface is obtained.
3. Calculate the partial derivative. According to the parameterized form of the surface, the partial derivative of the vector field of each point on the surface is calculated.
4. Calculate divergence and curl. According to the definitions of divergence and curl, the divergence and curl of vector field at each point on the surface are calculated by using partial derivatives.
It should be noted that it is a complex process to calculate the divergence and curl of vector field on a surface, which requires some knowledge of differential geometry and mathematical skills. In practical problems, it may be necessary to use computer software or numerical methods for calculation.