How Do I Resolve Cyclical Issues in Integration by Parts for Laplace Transforms?

In summary, the conversation mainly centers around two inner product questions and how to solve them. The first question involves demonstrating that <cos(mx), sin(nx)> = 0, <cos(mx), cos(mx)> = 1, and <sin(nx), sin(nx)> = 1. The second question involves verifying that the "1 norm" is a norm on Rn, which is done by applying integration by parts or looking it up in a table of integrals. There is also a brief mention of an unrelated issue regarding Laplace transforms and integration by parts.
  • #1
Dustinsfl
2,281
5
Last two inner product questions.

The first one I am little confused on and the second one I don't know what to do.

See worked attached.
 

Attachments

  • Untitled (1).pdf
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  • #2
What are you confused about?
 
  • #3
I don't know how to do it so I am confused on what to do.
 
  • #4
If two vectors are orthogonal, what does that mean about their inner product? And what does it mean to say a vector is a unit vector?
 
  • #5
Dustinsfl said:
Last two inner product questions.

The first one I am little confused on and the second one I don't know what to do.

See worked attached.

For the first one, show that <cos(mx), sin(nx)> = 0. Also show that <cos(mx), cos(mx)> = 1 and that <sin(nx), sin(nx)> = 1.

For the second one, you're showing that the "1 norm" is a norm on Rn. Verify that the formula satisfies all the defining properties of a norm: positive definiteness, etc. BTW, this norm simply adds the absolute values of the coordinates of a vector. For example, in R3, ||<2, -1, 3>||1 = |2| + |-1| + |3| = 6.
 
  • #6
How can I integrate cos(mx)*sin(nx) when they don't have the same angle?
 
  • #7
Use a trig identity to rewrite the product of a sine and a cosine.
 
  • #8
For the second problem, I don't know how to start the proof.
 
  • #9
Integration by parts twice would probably work, or you could look in a table of integrals. Since this isn't about learning to integrate, but is instead and application of integration, it seems reasonable to me to look it up in a table of integrals.
 
  • #10
On the integration by parts comment, I have an unrelated issue. I was doing a Laplace Transform of e^(-st+t)*sin(t) but after doing integration by parts, I had a form of the original with a 1/(s-1) time the integral so I can't subtract across to the other side. What can I do there since if I keep going it will an endless cyclical cycle?
 
  • #11
I have attached a update that show cos(mx) and sin(nx) are orthogonal; however, showing cos(mx) cos(mx) equals 1 isn't quite working.
 

Attachments

  • Untitled (2).pdf
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  • #12
It didn't say it explicitly in the problem statement, but you need to assume m and n are integers. What is sine evaluated at any multiple of 2π?
 
  • #13
If they are integers, then of course sin goes to zero.
 
  • #14
Dustinsfl said:
On the integration by parts comment, I have an unrelated issue. I was doing a Laplace Transform of e^(-st+t)*sin(t) but after doing integration by parts, I had a form of the original with a 1/(s-1) time the integral so I can't subtract across to the other side. What can I do there since if I keep going it will an endless cyclical cycle?
Can you elaborate on this a bit more? You haven't said quite enough so that I'm not sure I understand what you're talking about. If you have a given integral on one side, and 1/(s - 1) times the same integral on the other side, add -1/(s - 1) times the integral to both sides, and then combine the two integrals using the rules of plain old fractions. At that point you can solve for the integral algebraically.

If that's not what you're talking about, set me straight, please.
 

FAQ: How Do I Resolve Cyclical Issues in Integration by Parts for Laplace Transforms?

What is the inner product of two vectors?

The inner product of two vectors is a mathematical operation that takes two vectors and produces a scalar. It is also known as the dot product and is calculated by multiplying the corresponding components of the two vectors and adding the results together.

How is the inner product used in mathematics?

The inner product is used in mathematics to determine the angle between two vectors, to project one vector onto another, and to calculate the magnitude (or length) of a vector.

What is the geometric interpretation of the inner product?

The inner product can be interpreted geometrically as the product of the lengths of two vectors and the cosine of the angle between them. This means that the inner product is positive if the angle between the vectors is acute, negative if the angle is obtuse, and zero if the vectors are perpendicular.

What is the difference between the inner product and the cross product?

The inner product is a scalar value, while the cross product produces a vector. Additionally, the inner product is commutative (a·b = b·a), while the cross product is anti-commutative (a×b = -b×a). The inner product is also used for calculating the angle between two vectors, while the cross product is used for calculating the area of a parallelogram formed by two vectors.

How is the inner product used in physics?

In physics, the inner product is used to calculate work, energy, and power in a system. It is also used in the study of waves and harmonic motion, as well as in quantum mechanics to calculate probabilities and expectation values.

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