Vector calculus- region-density-mass

In summary: The Attempt at a Solution I am not sure what equation to use for the volume[/B]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,
  • #1

Homework Statement


https://www.physicsforums.com/attachments/229290
upload_2018-8-15_16-23-12.png

Homework Equations



upload_2018-8-15_16-24-25.png

The Attempt at a Solution


39245723_483901692074113_9059923629021593600_n.jpg

I am not sure what equation to use for the volume[/B]
 

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  • #2
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.
 
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  • #3
Chandra Prayaga said:
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.
39177607_1094553904033685_3707489651934625792_n.jpg

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
I guess it is just the triangle, because we have
upload_2018-8-15_22-58-21.png

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Is my final equation ok?
 

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  • #5
Jozefina Gramatikova said:
I was wondering if I need the whole rectangle as a region or just the purple triangle that I sketched there?
Your sketch is nowhere close to being right. It is a three dimensional solid.
What is the range of z? For some arbitrary z in that range, what does the XY lamina look like?
 

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with functions and operations involving vectors, which are quantities that have both magnitude and direction. It is used to study and analyze physical quantities that vary in space and time, such as velocity, force, and electric fields.

2. How is vector calculus related to region, density, and mass?

In vector calculus, region, density, and mass are often used to represent physical quantities in three-dimensional space. For example, the mass of an object can be represented as a density function over a particular region, and vector calculus can be used to find the total mass by integrating over that region.

3. What is the difference between scalar and vector fields in vector calculus?

A scalar field is a function that assigns a scalar value (such as temperature or pressure) to each point in space, while a vector field is a function that assigns a vector (such as velocity or force) to each point in space. In vector calculus, we use different operations and theorems for scalar and vector fields.

4. What are some important applications of vector calculus?

Vector calculus has many practical applications in physics, engineering, and other fields. Some examples include using vector calculus to study fluid flow, electromagnetics, and motion in three-dimensional space. It is also used in computer graphics and machine learning algorithms.

5. How can I improve my understanding of vector calculus?

To improve your understanding of vector calculus, it is important to have a strong foundation in calculus, linear algebra, and trigonometry. Practice solving problems and applying concepts to real-world situations. There are also many online resources and textbooks available for further learning and practice.

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