Surface integral problem formula question

In summary, the conversation discusses the use of a formula for calculating surface integrals, which involves taking into account the parametric equations for a surface. It is mentioned that any surface with the equation z = g(x,y) can be considered as a parametric surface with the parametric equations x=x, y=y, and z=g(x,y). This allows for the integration to be done over a two-dimensional region in the x-y plane instead of a three-dimensional surface.
  • #1
flyingpig
2,579
1

Homework Statement



My book says proves this formula

[tex]\iint_S f(x,y,z) dS = \iint_D f(x,y,g(x,y)) \sqrt{\left (\frac{\partial z}{\partial x} \right )^2 + \left (\frac{\partial z}{\partial y} \right )^2 + 1 } \;dA[/tex]

Any surface with equation z = g(x,y) can be regarded as a parametric surface with parametric equations

x = x

y = y

z = g(x,y)

rx = i + gx k


ry = j + gy k


|rx x ry| = [tex]\sqrt{\left (\frac{\partial z}{\partial x} \right )^2 + \left (\frac{\partial z}{\partial y} \right )^2 + 1 }[/tex]

Thus


[tex]\iint_S f(x,y,z) dS = \iint_D f(x,y,g(x,y)) \sqrt{\left (\frac{\partial z}{\partial x} \right )^2 + \left (\frac{\partial z}{\partial y} \right )^2 + 1 } \;dA[/tex]


Question

How do they know that every parametrization falls nicely as x = x and y = y?
 
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  • #2
In general, the surface is w = f(x, y, z), but if we're given that z is a function of x and y (i.e., z = g(x, y)), then we have w = f(x, y, g(x, y)). And instead of integrating over some surface S in three-dimensional space, we can integrate over a two-dimensional region in the x-y plane, D. I think that's all they're saying.
 

1. What is a surface integral?

A surface integral is a mathematical concept used in multivariable calculus to calculate the total value of a function over a given surface. It involves breaking down the surface into small, infinitesimal pieces and summing up the contributions of each piece to the overall value of the function.

2. How is a surface integral different from a regular integral?

A regular integral calculates the area under a curve in two-dimensional space, while a surface integral calculates the area of a three-dimensional surface. In a regular integral, the independent variable is typically a single variable, while in a surface integral, the independent variables are usually two parameters that define the surface.

3. What is the formula for solving a surface integral?

The formula for a surface integral depends on the type of surface and the function being integrated. For a smooth surface, the formula is given by ∫∫f(x,y)√(1+(∂z/∂x)²+(∂z/∂y)²)dA, where the limits of integration are determined by the boundaries of the surface.

4. What are some real-world applications of surface integrals?

Surface integrals have various applications in physics, engineering, and other scientific fields. They can be used to calculate the flux of a vector field through a surface, the work done by a force on a moving object, or the mass of a three-dimensional object with varying density.

5. How can I improve my understanding of surface integrals?

The best way to improve your understanding of surface integrals is to practice solving problems and working through examples. You can also seek out resources such as textbooks, online tutorials, or ask for help from a tutor or instructor if you are struggling with a particular concept or formula.

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