- #1
Jozefina Gramatikova
- 62
- 9
Vector calculus- region-density-mass
- Thread starter Jozefina Gramatikova
- Start date
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Homework Help Overview
The discussion revolves around a problem in vector calculus related to calculating mass based on a density function that varies with position. Participants are examining the appropriate equations and methods to determine the mass of a three-dimensional region defined by specific bounds in x, y, and z.
Discussion Character
- Exploratory, Conceptual clarification, Assumption checking
Approaches and Questions Raised
- Participants are questioning the adequacy of the original equations presented for calculating mass, particularly in the context of a variable density function. There is a focus on understanding the geometric representation of the region in question and whether the entire rectangular region or just a specific triangular section should be considered.
Discussion Status
Some guidance has been offered regarding the need to visualize the region and reconsider the equations used for variable density. Multiple interpretations of the region's shape are being explored, and there is an ongoing dialogue about the correct approach to take.
Contextual Notes
Participants are discussing the implications of the density function being dependent on the z-coordinate and the necessity of integrating over the defined volume to find total mass. There is a mention of the original poster's sketch being unclear, which may affect the understanding of the problem.
- #2
Chandra Prayaga
Science Advisor
- 652
- 150
The attempt that you posted is too small and I cannot click on it to get a better view. But from what I can see, I suggest the following.
1. Understand, from the ranges of x, y and z given by you, the shape of the region you are interested in. In other words, answer the first question and sketch the region. In your graph, you only showed a few isolated points. You did not sketch the region in space.
2. The relevant equation posted by you is not adequate. Density = mass / volume is good only if the density is constant. In your case, the density depends on position (your equation: ρ = 1 + z). Then you must use the differential form:
dm = ρ dV,
and integrate this to get the total mass. It will be a triple integral.
1. Understand, from the ranges of x, y and z given by you, the shape of the region you are interested in. In other words, answer the first question and sketch the region. In your graph, you only showed a few isolated points. You did not sketch the region in space.
2. The relevant equation posted by you is not adequate. Density = mass / volume is good only if the density is constant. In your case, the density depends on position (your equation: ρ = 1 + z). Then you must use the differential form:
dm = ρ dV,
and integrate this to get the total mass. It will be a triple integral.
- #3
Jozefina Gramatikova
- 62
- 9
Chandra Prayaga said:
I hope you can see the picture better now. I was wondering if I need the whole rectangle as a region or just the purple triangle that I sketched there?
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- #4
- #5
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Your sketch is nowhere close to being right. It is a three dimensional solid.Jozefina Gramatikova said:
What is the range of z? For some arbitrary z in that range, what does the XY lamina look like?
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