CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND SURFACE INTEGRALS P(x,y) xy-plane, dxdy If is a point in the the element of area has a vector area dx dynˆ ˆdx ˆdy   d S kdxdy Z p X Y O dx Where is a vector normal to the plane made by and directions. kˆ iˆ ˆj dSnˆ Thus we write . On writing we have d S  dxdynˆ dS  dxdynˆ d S  dsnˆ S ds where is a general surface in space, and is an element of ds S surface. We also know from the previous chapter that the unit normal to a surface S is nˆ given by grad S   nˆ grad S  Hence d S ds For a vector field and a scalar field, the integrals  F S S S S S S are examples of surface integrals. To evaluate surface integrals, it is convenient to express them as double integrals taken over the projected area of the surface S on one of the
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS SURFACE INTEGRALS If P(x,y) is a point in the xy-plane, the element of area dxdy has a vector area d S  ˆdx ˆdy  kdxdy Z dx dynˆ X O dx p Y Where kˆ is a vector normal to the plane made by iˆ and ˆj directions. Thus we write d S  dxdynˆ . On writing dS  dxdynˆ we have d S  dsnˆ dSnˆ where S is a general surface in space, and ds is an element of S ds surface. We also know from the previous chapter that the unit normal nˆ to a surface S is given by nˆ  grad S grad S  Hence d S  ds For F a vector field and  a scalar field, the integrals S S S S S S are examples of surface integrals. To evaluate surface integrals, it is convenient to express them as double integrals taken over the projected area of the surface S on one of the