Calculate Line Integral for a Function on a Level Surface | Homework Equations

In summary, the conversation discusses finding the integral grad f dot dr when given a function of three variables and an oriented curve lying on the level surface f(x,y,z) = 5. The solution involves taking the difference between the values of the function at two endpoints on the curve, which in this case is 5. It is also noted that grad f is not equal to (0,0,0) because f(x,y,z) = 5 only on the level surface, not everywhere.
  • #1
joemama69
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0

Homework Statement



Let f(x,y,z) be a function of three variables. Suppose that C is an oriented curve lying on the level surface f(x,y,z) = 5. Find the integral grad f dot dr.

Homework Equations





The Attempt at a Solution



integral grad f dot dr = integral f(q) - f(p)

not sure what to do
 
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  • #2
If a point p is on the level surface f(x,y,z) = 5, what does that tell you about f(p)?
 
  • #3
If the curve you are integrating over lies on the surface f(x,y,z)=5, then surely the endpoints p and q do as well right?...So f(p)=___? and f(q)=___?
 
  • #4
f(P) - f(Q) = 5 - 5 = 0
 
  • #5
is that correct
 
  • #6
It's correct.
 
  • #7
If f(x,y,z) = 5, how is grad f not equal to (0,0,0)? Jeez, today's just not my day.
 
  • #8
f(x,y,z) = 5 on the level surface, not everywhere. If it was equal to 5 everywhere, grad f would be zero.
 
  • #9
Oh yeah, duh. I think I need to drink more water or something.
 

1. What is a line integral?

A line integral is a type of integral in calculus that is used to calculate the total value of a function along a given curve or line. It takes into account both the distance and direction of the curve to determine the total value.

2. How is a line integral calculated?

To calculate a line integral for a function on a level surface, you first need to parameterize the curve or line. This means expressing the x, y, and z coordinates in terms of a single variable, usually t. Then, you can plug this parameterization into the integral formula and calculate the integral using techniques from calculus.

3. What is a level surface?

A level surface is a surface in 3-dimensional space where every point on the surface has the same value for a given function. This can be visualized as a flat plane or curved surface where the height or depth is constant across the entire surface.

4. Why is calculating a line integral on a level surface useful?

Calculating a line integral on a level surface can be useful in many areas of science and engineering, such as calculating work done by a force, finding the mass of a 3-dimensional object, or determining the circulation of a vector field. It allows for the evaluation of a function along a specific path or curve, rather than just at specific points.

5. Are there any limitations to calculating a line integral on a level surface?

Yes, there are some limitations to calculating a line integral on a level surface. One limitation is that the function must be continuous and differentiable along the curve. Additionally, the curve must be a closed loop or have a defined start and end point. In some cases, the calculation of a line integral on a level surface may also require advanced mathematical techniques and can be time-consuming.

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