MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms V4 Green s Theorem in Normal Form 1 Green s theorem for flux Let F M i N j represent a two dimensional flow field and C a simple closed curve positively oriented with interior R According to the previous section 1 flux of F across C Notice that since the normal vector points outwards away from R the flux is positive where the flow is out of R flow into R counts as negative flux We now apply Green s theorem to the line integral in 1 first we write the integral in standard form dx first then dy This gives us Green s theorem in the normal form Mathematically this is the same theorem as the tangential form of Green s theorem all we have done is to juggle the symbols M and N around changing the sign of one of them What is different is the physical interpretation The left side represents the flux of F across the closed curve C What does the right side represent 2 The two dimensional divergence Once again let F M i N j We give a name to and a notation for the integrand of the double integral on the right of 2 dM dN div F dx dy the divergence of F Evidently div F is a scalar function of two variables To get a t its physical meaning look at the small rectangle pictured If F is continuously differentiable then div F is a continuous function which is therefore approximately constant if the rectangle is small enough We apply 2 to the rectangle the double integral is approximated by a product since the integrand is approximately constant 4 flux across sides of rectangle z E Y AA AA area of rectangle Because of the importance of this approximate relation we give a more direct derivation of it which doesn t use Green s theorem The reasoning which follows is widely used in mathematical modeling of physical problems V VECTOR INTEGRAL CALCULUS 2 Consider the small rectangle shown We calculate an approximate value for the flux over each of its sides flux across top flux across bottom F x y Ay j Ax N x y F x y j Ax Ay AX N x y Ax Y AY adding these up total flux across top and bottom x y A N X Y AX g i M x y AX By similar reasoning applied to the two sides total flux across left and right sides AX y M x y 2 AX AY Adding up the flux over the four sides we get 4 again total flux over four sides of the rectangle dM dN Continuing our search for a physical meaning for the divergence if the total flux over the sides of the small rectangle is positive this means there is a net flow o u t of the rectangle According to conservation of matter the only way this can happen is if there is a source adding fluid directly to the rectangle If the flow is taking place in a shallow tank of uniform depth such a source can be visualized as someone standing over the tank pouring fluid directly into the rectangle Similarly a net flow i n t o the rectangle implies there is a sink withdrawing fluid from the rectangle It is best to think of such a sink as a negative source The net rate positive or negative at which fluid is added directly t o the rectangle from above may be called the source rate for the rectangle Thus since matter is conserved flux over sides of rectangle source rate for the rectangle combining this with 4 shows that source rate for the rectangle z dM dN We now divide by AA and pass to the limit getting by definition the source rate at x y div F The definition of the double integral as the limit of a sum shows in the usual way now that source rate for R L div F d These two relations 6 and 7 interpret the divergence physically for a flow field and they interpret also Green s theorem in the normal form total flux across C source rate for R X x Ax V4 GREEN S THEOREM IN NORMAL FORM 3 Since Green s theorem is a mathematical theorem one might think we have proved the law of conservation of matter This is not so since this law was needed for our interpretation of div F as the source rate at x y We give side by side the two forms of Green s theorem first in the vector form then in the differential form used when calculations are to be done Tangential form Normal form flux of F across C work by F around C source rate for R 3 An interpretation for curl F The function curl F can be thought of as measuring the rotational tendency of the vector field either as a force field or a velocity field F will make a test object placed at a point Po spin about a vertical axis i e one in the k direction and the angular velocity of the spin will be proportional to curl F O To see this for the velocity field v of a flowing liquid place a paddle wheel of radius a so its center is at xo yo and its axis is vertical We ask how rapidly the flow spins the wheel If the wheel had only one blade the velocity of the blade would be F t the component of the flow velocity vector F perpendicular to the blade i e tangent to the circle of radius a traced out by the blade x d s 6 yo I paddlewheel top view Since F t is not constant along this circle if the wheel had only one blade it would spin around at an uneven rate But if the wheel has many blades this unevenness will be averaged out and it will spin around at approximately the average value of the tangential velocity F t over the circle Like the average value of any function defined along a curve this average tangential velocity can be found by integrating F t over the circle and dividing by the length of the circle Thus F t ds speed of blade 1 L curl F o dx dy h F dr by Green s theorem where curl F o is the value of the function curl F at xo y o The justification for the last approximation is that if the circle formed by the paddlewheel is small then curl F has approximately the value curl F o over the interior R of the circle so that multiplying this constant value by the area nu2 of R should give approximately the value of the double integral V VECTOR INTEGRAL CALCULUS 4 From 8 we get for the tangential speed of the paddlewheel tangential speed z a 2 curl F …
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