How to prove this corollary in Line Integral using Riemann integral

In summary, we can prove that for a smooth curve $C$ with arc length $L$ and a vector field $f(x,y) = P(x,y)i + Q(x,y)j$ with a maximum magnitude of $M$ on $C$, the integral of $f$ along $C$ is bounded by $ML$. This can be shown by using the hint and the fact that the Riemann Integral of a function is bounded by the integral of its absolute value.
  • #1
WMDhamnekar
MHB
376
28
. Let C be a smooth curve with arc length L, and suppose that f(x, y) = P(x, y)i +Q(x, y)j is a vector field such that $|| f|(x,y) || \leq M $ for all (x,y) on C. Show that $\left\vert\displaystyle\int_C f \cdot dr \right\vert \leq ML $
Hint: Recall that $\left\vert\displaystyle\int_a^b g(x) dx\right\vert \leq \displaystyle\int_a^b |g(x) | dx $ for Riemann Integrals.

Now, how to prove this corollary using this hint?
 
Physics news on Phys.org
  • #2
Assuming $C$ is parametrized by $r(t)$, $a\le t \le b$, write $$\int_C f\cdot dr = \int_a^b f(r(t))\cdot r'(t)\, dt$$ By the hint, $\int_C f\cdot dr$ is bounded by $\int_a^b |f(r(t))\cdot r'(t)|\, dt$. Since $|f(r(t))\cdot r'(t)| \le |f(r(t))| |r'(t)| \le M |r'(t)|$ for all $t$, we deduce $\left|\int_C f\cdot dr\right| \le M \int_a^b |r'(t)|\, dt = ML$.
 

Related to How to prove this corollary in Line Integral using Riemann integral

1. How do I use Riemann integrals to prove a corollary in line integrals?

To prove a corollary in line integrals using Riemann integrals, you will need to use the Fundamental Theorem of Calculus for Line Integrals. This theorem states that the line integral of a vector field over a curve can be calculated by evaluating the corresponding Riemann integral. You will also need to use the definition of a line integral and properties of Riemann integrals.

2. Can I use any Riemann integrable function to prove the corollary in line integrals?

No, you cannot use any Riemann integrable function to prove the corollary in line integrals. The function must be continuous on the curve and satisfy certain conditions, such as being bounded and having a finite number of discontinuities. It is important to carefully choose the function to ensure that the corollary can be proven.

3. What is the relationship between line integrals and Riemann integrals?

Line integrals and Riemann integrals are closely related. In fact, the line integral of a vector field over a curve can be calculated by evaluating the corresponding Riemann integral. This is known as the Fundamental Theorem of Calculus for Line Integrals. Riemann integrals are also used to prove corollaries in line integrals.

4. Is it necessary to use Riemann integrals to prove a corollary in line integrals?

Yes, it is necessary to use Riemann integrals to prove a corollary in line integrals. Riemann integrals are essential in calculating line integrals and are also used to prove theorems and corollaries related to line integrals. They provide a rigorous mathematical framework for understanding and solving problems involving line integrals.

5. Are there any alternative methods for proving a corollary in line integrals?

Yes, there are alternative methods for proving a corollary in line integrals. One possible method is to use Green's Theorem, which relates a line integral over a curve to a double integral over a region in the plane. Another method is to use Stokes' Theorem, which relates a line integral over a curve to a surface integral over a surface bounded by the curve. However, these methods may not always be applicable and may require additional conditions or assumptions.

Similar threads

  • Calculus
Replies
6
Views
1K
Replies
16
Views
3K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
6
Views
1K
Replies
3
Views
1K
Replies
1
Views
2K
Replies
0
Views
446
  • Calculus
Replies
11
Views
2K
Replies
20
Views
2K
Back
Top