Changes

Let <math>x=x(u,v),y=y(u,v),z=z(u,v)  </math> be a parametric description of <math>S </math>, so that ''S'' will be defined as the image under these functions these three functions ''x,y,z'' of some region R of the ''uv'' plane.   

Let <math>x=x(u,v),y=y(u,v),z=z(u,v)  </math> be a parametric description of <math>S </math>, so that ''S'' will be defined as the image under these functions these three functions ''x,y,z'' of some region R of the ''uv'' plane.   

As before, the trick to finding a convenient formula for a surface integral is finding a formula for that surface area element, ''dA''.  What does such a surface element even look like?

You'll remember that in our second lecture, we defined the cross product of two vectors <math>\vec{u}\times\vec{v}</math> as having length equal to the area of the parallelogram bordered on two sides by <math>\vec{u}</math>  and <math>\vec{v}</math>, and direction perpendicular to either vector, with a right handed orientation.  We can use this to our advantage now - all we need to do is figure out two vectors which will border a surface area element, say, <math>\vec{u}, \vec{v}</math>, and we'll be able to compute the surface integral as

You'll remember that in our second lecture, we defined the cross product of two vectors <math>\vec{u}\times\vec{w}</math> as having length equal to the area of the parallelogram bordered on two sides by <math>\vec{u}</math> and <math>\vec{w}</math>, and direction perpendicular to either vector, with a right handed orientation.

<math>\iint_S f(x,y,z)dA = \iint_R f(x(u,v),y(u,v),z(u,v)) |\vec{u}\times\vec{v}| dudv </math>

{{MVCalc}}

{{MVCalc}}