Question

Define a surface integral of the scalar function \(f\) over a surface \(z=g(x, y)\). Explain how to evaluate the surface inteoral

Short answer

The surface integral of the scalar function 'f' over a surface 'z = g(x, y)' is given by the formula: \[ \iint_S f\, dS = \iint_D f(g(x, y))\sqrt{1 + (\frac{\partial g}{\partial x})^2 + (\frac{\partial g}{\partial y})^2}\, dx\, dy \], where 'D' is the projection of 'S' onto the xy-plane.

Step-by-step solution

Understand Surface Integral

A surface integral is a generalization of line integrals, extending the idea to integration over surfaces. In this case, for a scalar function 'f' defined over some surface 'S', the surface integral calculates the sum of the values of 'f' over 'S'.

Write the Formula of Surface Integral

For a function 'f' defined over a surface 'z = g(x,y)', the formula of the surface integral is given as: \[ \iint_S f\, dS = \iint_D f(g(x, y))\sqrt{1 + (\frac{\partial g}{\partial x})^2 + (\frac{\partial g}{\partial y})^2}\, dx\, dy \] where 'D' is the projection of 'S' onto the xy-plane.

Breakdown the Formula

The function 'f(g(x, y))' represents the function 'f' defined over the surface 'S'. The square root term, 'sqrt(1 + (\frac{\partial g}{\partial x})^2 + (\frac{\partial g}{\partial y})^2)', is the magnitude of the normal vector to the surface 'S', derived from the gradient of 'g'.

Calculate the Surface Integral

To evaluate the integral, firstly compute the partial derivatives of 'g' with respect to 'x' and 'y'. Secondly, substitute the function and the partial derivatives into the formula. Lastly, make sure to find the limits of integration 'D' that cover the entire surface 'S'.