Algebra: How do I derive this equation given two other equations?

In summary, the conversation discusses the equations shown in an image and the difficulties the speaker had in deriving equation 5. They realized their mistake in assuming the wrong form for the equation and were able to find the correct solution. The speaker also provides a link to the source of the equations and clarifies their notation for r prime in their work. Overall, the conversation ends with the speaker thanking the other person for their help and considering the issue resolved.
  • #1
dsilvas
7
4
Homework Statement
Derive eq.(5) and eq.(6) from eq.(3) and (4)
Relevant Equations
Unsure how to type the equations correctly in this text box (no formating options). They are below.
hw9.JPG

This image shows the equations.
I managed to almost get equation 5, but my partial derivative is not squared but instead multiplied by mu, and also I don't have a factor of 1/2.
Here is an image of the work I have. I'm sorry for any sloppiness. I tried to be as concise as possible when writing it down but it should be pretty straightforward. Essentially the only way my final equation could be equal to eq.(5) is if mu is equal 1/2 of its partial derivative with respect to r.
hw9_mywork.jpg

If it helps, here is the link to the paper that these equations are coming from. It is page 2.
https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/98RS01523
I didn't write it out explicitly, so to elaborate, r prime in my work is equal to the full derivative of r with respect to theta. I was able to derive eq. (6) from eq. (4) and that has helped me get closer to deriving eq. (5) but I'm not quite there and I'm stuck.
 

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  • #2
Hi.
In the last line in your notebook substitute
[tex]\mu \frac{d\mu}{dr}=\frac{1}{2}\frac{d\mu^2}{d\mu}\frac{d\mu}{dr}=\frac{1}{2}\frac{d\mu^2}{dr}[/tex]
 
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  • #3
mitochan said:
Hi.
In the last line in your notebook substitute
[tex]\mu \frac{d\mu}{dr}=\frac{1}{2}\frac{d\mu}{dr}[/tex]

Oh gosh. I think I realize my mistake now, thank you.
I assumed in eq 5 that it was [tex]{(\frac{d\mu}{dr})}^2[/tex] when in reality it was [tex]\frac{d(\mu^2)}{dr}[/tex]
I just can't read. Thank you. I didn't realize it was that simple.

I guess this question can be closed now!
 
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Related to Algebra: How do I derive this equation given two other equations?

1. How do I derive an equation by combining two other equations?

To derive an equation using two other equations, you can use the substitution method or the elimination method. In the substitution method, you solve one of the equations for a variable and then substitute that value into the other equation. In the elimination method, you manipulate the equations to eliminate one variable, and then solve for the remaining variable.

2. What is the process for solving a system of equations?

To solve a system of equations, you need to first identify the type of system (linear, quadratic, etc.). Then, you can use methods such as substitution, elimination, or graphing to find the values of the variables that satisfy both equations. Finally, you can check your solutions by plugging them back into the original equations.

3. Can I use algebra to solve real-world problems?

Yes, algebra is a powerful tool for solving real-world problems. It allows you to represent and manipulate relationships between variables and use them to find unknown quantities. Many fields, such as engineering, physics, and economics, rely on algebra to solve complex problems.

4. How do I know if a system of equations has a solution?

A system of equations has a solution when the equations intersect at a single point, meaning that the values of the variables satisfy both equations. This can be determined by graphing the equations or by solving them using algebraic methods. If the equations do not intersect, then there is no solution.

5. What is the importance of understanding algebra in scientific fields?

Algebra is a fundamental tool in scientific fields as it allows you to model and analyze complex systems and relationships. It is used to solve equations, make predictions, and understand patterns and trends in data. Without a strong understanding of algebra, it would be difficult to make progress in many scientific disciplines.

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