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[TabrisRyu]

...

Hello! Thanks for replying! Maybe I can help with the definitions a bit.

And this is how I did the last part:

The problem is asking for the n derivative of f(x) , (x-1)^-(1 underscore) is f(x) in this case I guess. I don't think it is related to the laplacian operator, sadly.

Regarding the notation, I have found this: Pochhammer Symbol. This is called falling factorial and rising factorial : P


I'm really grateful for your help! This is giving me nightmares

 

[kgtrep]

Regarding the notation, I have found this: Pochhammer Symbol. This is called falling factorial and rising factorial.

Thanks, I'll think about it when I get tired with work today. In your proof that you posted above, why does property (3) imply that, (using Pochhammer symbols)

(x - 1)_(-1 - n) = 1 / (x)^(1 + n)?

Wouldn't it be,

(x - 1)_(-1 - n) = 1 / (x - 1 + 1 + n)^(1 + n) = 1 / (x + n)^(1 + n)?

Edit: Delta seems analogous to the usual differentiation d/dx, and the rising/falling exponents seem to follow the usual exponentiation rules. I feel like, as a result, property (3) should say x_(-n) = 1 / (x)^n and your previous problem,

Delta^n (x - 1)_(-1) = (-1)^n * n! / (x - 1)^(n + 1).

 

[TabrisRyu]

Thanks, I'll think about it when I get tired with work today. In your proof that you posted above, why does property (3) imply that, (using Pochhammer symbols)

(x - 1)_(-1 - n) = 1 / (x)^(1 + n)?

Wouldn't it be,

(x - 1)_(-1 - n) = 1 / (x - 1 + 1 + n)^(1 + n) = 1 / (x + n)^(1 + n)?

Edit: Delta seems analogous to the usual differentiation d/dx, and the rising/falling exponents seem to follow the usual exponentiation rules. I feel like, as a result, property (3) should say x_(-n) = 1 / (x)^n and your previous problem,

Delta^n (x - 1)_(-1) = (-1)^n * n! / (x - 1)^(n + 1).

Wow I'm really an idiot, when I wrote it again so I could take the picture I misread (3)

(3) :

(x)^-n = 1 / ( x + 1)^

x to the -n underscore equals 1 / (x+1) to the n upperscore.

You can also express a falling factorial as a rising factorial and vice-versa:

x^ = (( x + n - 1))^n

(x)^n = (x-n+1)^

 

[kgtrep]

(x)^-n = 1 / ( x + 1)^

x to the -n underscore equals 1 / (x+1) to the n upperscore.

That's even better.

The first line is because, with usual exponents,

(-1)^k = (-1)^n * (-1)^(k - n) = (-1)^n * (-1)^(n - k).

I was trying to intentionally introduce the term (-1)^(n - k), so that that the summation equals to Delta^n f(x), where f(x) = 1/x.

 

[TabrisRyu]

That's even better.

The first line is because, with usual exponents,

(-1)^k = (-1)^n * (-1)^(k - n) = (-1)^n * (-1)^(n - k).

I was trying to intentionally introduce the term (-1)^(n - k), so that that the summation equals to Delta^n f(x), where f(x) = 1/x.

This is beautiful... Thank you very much!!!

What I didn't guess was that you could get (-1)^(n-k) with the (-1)^k. This was the first step and I could not see it...

It is correct to say that in the second line you use x^(-1) because (x+k) is in the denominator in the expression?

Then what you do is following (3) you know that

x^(-1) expressed in factorial falling is (x-1)^-1underscore, then you apply the definition gotten in the last exercise.. I see it.

You sir are a genius!

Now I'll do the next one

 

[Majestad]

This seems like an easy problem for somebody more experienced, but like I said in another thread, the teacher I got is absolutely god awful at explaining things so I'm at a complete lost here. Any help is greatly appreciated.

This is a pre-calculus problem:

Suppose a car travels at a constant rate of 55mph for 2 hours and travels at 45mph thereafter. Show that distance is a function of time, and find the rule of function.

If anybody has patience, can you explain this problem to me in detail? Thanks a lot!

 

[kgtrep]

This is beautiful... Thank you very much!!!

What I didn't guess was that you could get (-1)^(n-k) with the (-1)^k. This was the first step and I could not see it...

It is correct to say that in the second line you use x^(-1) because (x+k) is in the denominator in the expression?

Then what you do is following (3) you know that

x^(-1) expressed in factorial falling is (x-1)^-1underscore, then you apply the definition gotten in the last exercise.. I see it.

You sir are a genius!

Now I'll do the next one

Thanks, but it was really because your list of definitions was clear. I saw from your definition of Delta^n f(x) how we should manipulate the starting expression, so that we can write down the function f; that gives us the second line.

 

[FullmetalPain]

Does anyone have a paid course hero account? I need help with this matlab assignment and I would appreciate if someone could download these documents for me.

https://www.coursehero.com/file/7985780/MATLAB-Project/

 

[MoG EclipsE]

Since we are to use first-order forward integration in time, we have,

We know T(x, y; t = 0), as it's given by the initial condition. In particular, the initial temperature is 10 degrees Celsius at the four points.

All that remains is to evaluate the time derivative (for convenience, I'll drop the argument t = 0) while using the central finite difference scheme to approximate the Laplacian operator. Recall the 5-point stencil:

I've written the formula above in the usual notation; it looks complicated, but it's really not. The (x_i, y_j) is the point at which we want to evaluate the Laplacian. Central scheme says we can make a second-order approximation using 5 nearby points including itself. The -1 in the index means look to the left/below of this point, while +1 means look to the right/above of this point. h is the stencil size--assumed to be the same in x and y directions--and is equal to 1 m in your problem, so I'll omit this number below. (Make sure that the units are consistent for real problems.)

For example, according to the scheme, the Laplacian at point #1 is,

20 + 10 + 10 + 30 - 4(10) = 30,

while at point #2, it is,

10 + 50 + 10 + 30 - 4(10) = 60.

Now you have all the information needed to approximate the time derivative at t = 0 at the four points:

You would repeat the procedure above, and use the information from t = 0.1 to approximate the temperature at t = 0.2 at point #3. The problem doesn't explicitly say this, but the temperature on the boundary is fixed in time, so we do not need to update the temperature for the boundary points at t = 0.1.

Thank you again for your help, I really appreciate it!

 

[Charmicarmicat]

I have a simple question that I can't work out, and can't find an answer to online. So, a number to the power of 2, then squared, obviously cancels out and leaves the number. What happens when it's to the power of minus 2?

 

[Stumpokapow]

I have a simple question that I can't work out, and can't find an answer to online. So, a number to the power of 2, then squared, obviously cancels out and leaves the number. What happens when it's to the power of minus 2?

You're not using the word "squared" correctly.
"A number to the power of two" is "a number squared"
"A number to the power of two squared" is that number to the power of 2*2 = 4
"The square root of a number to the power of two" does "cancel out" and leave the number.

The basic algebraic rules for exponents are:
- Multiplying two numbers of the same base means adding the exponents: (x^2)(x^3) = x^(2+3). (3^1)(3^1) = 3^(1+1) = 9.
- Raising a base to the power of an exponent and then raising the resulting number to the power of an exponent means multiplying the exponents: (x^2)^3 = x^(2*3) = x^6. (3^2)^5 = (3^10) = 59,049.
- Negative exponents can be treated by taking the inverse of the base and making the exponent positive. x^-1 = 1/(x^1). 3^-2 = 1/(3^2) = 1/9.
- "Squared" means "to the power of two". "Square root" means "to the power of one half".
- Anything to the power of 0 = 1

What you appear to be saying in your first sentence:
(x^2)^(1/2) = x^1 = x

Some things you might be asking:
(x^-2)^(1/2) = x^-1 = 1/x
(x^2)^(-2) = x^-4 = 1/(x^4)

It would help if you wrote out what you're trying to solve.

 

[Devil Theory]

So I'm not very mathematically inclined, and have a quick question:

Let's say I'm a business, and I expect to acquire 5000 entities in a 12-month period. How do I figure out and estimate a gradual percentage increase per month? So 5000/12 is around 416, but I don't expect to gain 416 in month one or two, so how do I figure out a growth that will maybe start picking up pace around month 4-5, and eventually end at 5000?

 

[Stumpokapow]

So I'm not very mathematically inclined, and have a quick question:

Let's say I'm a business, and I expect to acquire 5000 entities in a 12-month period. How do I figure out and estimate a gradual percentage increase per month? So 5000/12 is around 416, but I don't expect to gain 416 in month one or two, so how do I figure out a growth that will maybe start picking up pace around month 4-5, and eventually end at 5000?

Basically you want a sequence of 12 integers who sum to 5000. There are infinitely many such lists. You don't seem to say if the sequence is linear, geometric, exponential, or what.

Provided you wish to grow uniformly and linearly
Month i = (i-1)/11 * 834

Month 1 = 0
Month 2 = 76 (for the sake of simplicity, I will round up on even months and round down on odd months)
Month 3 = 151
Month 4 = 228
Month 5 = 303
Month 6 = 380
Month 7 = 454
Month 8 = 531
Month 9 = 606
Month 10 = 683
Month 11 = 758
Month 12 = 834
5004 total.

Another option would be to ramp up gradually. Say you decide to acquire one the first month, and you want to know how much you want to grow to get to 5000.
5000 = sum(i = 1 to 12) 1 * growthRate^(i-1)
growthRate = 2.04
Month i = 2.04^(i-1)
Months (again, round up one month and down the next): 1, 2.04, 4.16, 8.48, 17.31, 35.33, 72.07, 147.03, 299.94, 611.88, 1248.25, 2546.43

Another option might be to start with a little bit. Say you decide to acquire 20 the first month:
(5000/20) = sum(i = 1 to 12) growthRate^(i-1)
growthRate = 1.495
Month i = 20 * 1.495^(i-1)
Months: 20, 29, 44, 66, 99, 149, 223, 333, 499, 746, 1115, 1667


So really we don't have enough information to give you an answer, because there are infinitely many answers. How slow do you want to start? How much do you want to ramp up?

 

[spelltropy]

trying to wrap this (i think simple) surface integral question:

I've got f(x,y,z)=2z, and I'm trying to calculate the surface integral f dS for the the region S given by the hemisphere x^2+y^2+z^2=a^2 and z>=0. I decided to convert to spherical coordinates and got 0<r<a, 0<theta<2pi, 0<phi<pi for my new limits.

So my new integral was these limits and the integral of 2r^3cos phi sin phi - problem is, I got 0 for this, and i'm not sure where my error is coming from. Any ideas?

 

[Leezard]

trying to wrap this (i think simple) surface integral question:

I've got f(x,y,z)=2z, and I'm trying to calculate the surface integral f dS for the the region S given by the hemisphere x^2+y^2+z^2=a^2 and z>=0. I decided to convert to spherical coordinates and got 0<r<a, 0<theta<2pi, 0<phi<pi for my new limits.

So my new integral was these limits and the integral of 2r^3cos phi sin phi - problem is, I got 0 for this, and i'm not sure where my error is coming from. Any ideas?

The condition Z >0 will affect your condition on phi or theta.

 

[spelltropy]

The condition Z >0 will affect your condition on phi or theta.

I was just being stupid, drew a diagram and decided pi = 90 degrees. Thanks!

 

[Frillen]

Can anyone help me solve this. Full solution please!

 

[Stumpokapow]

it's not clear to me what the ":" operator is there--possibly asking some sort of ratio?
what's the relationship of 6a - 6 to a - 1? take 6 out: 6(a-1) = 6a - 6
what's the relationship of 12b^2 : 4b^3? take 3/b out 3/b out: 3/b (4b^3) = 12b^2

so I would say the ratio or multiplier is 6/(3/b) = 6b/3 = 2b?

(6a-6)/(12b^2) = 2b * (a-1)/(4b^3)

is that what the question is looking for?

 

[Frillen]

it's not clear to me what the ":" operator is there--possibly asking some sort of ratio?
what's the relationship of 6a - 6 to a - 1? take 6 out: 6(a-1) = 6a - 6
what's the relationship of 12b^2 : 4b^3? take 3/b out 3/b out: 3/b (4b^3) = 12b^2

so I would say the ratio or multiplier is 6/(3/b) = 6b/3 = 2b?

(6a-6)/(12b^2) = 2b * (a-1)/(4b^3)

is that what the question is looking for?

Yes And they want the answer to be as easy as possible, which I think your solution is..?

 

[Hypertrooper]

I am lost with this exercise. I have to show that Phi is a isomorphism. Therefore I have to show that Phi is injective and surjective.
(gcd(m,n)) has more than the unification of , (m) because we know that there is p with the attribute p|m and p|n => p|r*m and p|r*n. But r*p is in (gcd(m,n)) as well. But does that help me? I am lost.

 

[kgtrep]

I am lost with this exercise. I have to show that Phi is a isomorphism. Therefore I have to show that Phi is injective and surjective.
(gcd(m,n)) has more than the unification of , (m) because we know that there is p with the attribute p|m and p|n => p|r*m and p|r*n. But r*p is in (gcd(m,n)) as well. But does that help me? I am lost.

Whew, that's beyond my level in algebra. Can you define for us, so that we are all on the same page and can bounce around ideas with you, the following terms:

natural projection
Hom_Z

I'm not sure how to make sense of your second equation with Phi. Is Phi mapping the quantity Hom_Z(...) -> Z/(...), f to the quantity p(f(1+ )), or are Phi and f two separate maps that you have to show are isomorphisms?

 

[Therion]

Whew, that's beyond my level in algebra. Can you define for us, so that we are all on the same page and can bounce around ideas with you, the following terms:

natural projection
Hom_Z

I'm not sure how to make sense of your second equation with Phi. Is Phi mapping the quantity Hom_Z(...) -> Z/(...), f to the quantity p(f(1+ )), or are Phi and f two separate maps that you have to show are isomorphisms?

While I don't have a solution for the OP, I can answer your questions at least:

The natural projection p is the homomorphism taking an element k+(m) in Z/(m) to k+(gcd) in Z/(gcd).

Hom_Z(...) is the group of Z-module homomorphisms (i.e. homomorphisms as abelian groups) from Z/ to Z/(m). So we are showing that Phi is an isomorphism taking the homomorphism f to the equivalence class p(f(1 + )).

 

[kgtrep]

While I don't have a solution for the OP, I can answer your questions at least:

The natural projection p is the homomorphism taking an element k+(m) in Z/(m) to k+(gcd) in Z/(gcd).

Hom_Z(...) is the group of Z-module homomorphisms (i.e. homomorphisms as abelian groups) from Z/ to Z/(m). So we are showing that Phi is an isomorphism taking the homomorphism f to the equivalence class p(f(1 + )).

Thanks!

 

[Skittles]

I despise this guy's take home quizzes.

For number 1, I really only need help with part a.

For number 2, need help setting up part a.
For number 3, feel like it's easier than i'm making it to be.

 

[kgtrep]

I despise this guy's take home quizzes.

For number 1, I really only need help with part a.
For number 2, need help setting up part a.
For number 3, feel like it's easier than i'm making it to be.

The quiz is actually fair and does test you guys on vector spaces in a good way. I think your professor wants you to understand that you can view polynomials as vectors (how?)

Since it's a quiz, I'll give a general guide.

1a. The image of a function is how the function acts upon an input.

2a. Given a vector space V, we say that U, a subset of V, is a subspace if it is a vector space by its own. How do we show that a space is a vector space? Is there a better way to show this if we know that U is a subset of V, which is already a vector space?

3. Recall that we can view polynomials as vectors.

 

[kgtrep]

I am lost with this exercise. I have to show that Phi is a isomorphism. Therefore I have to show that Phi is injective and surjective.
(gcd(m,n)) has more than the unification of , (m) because we know that there is p with the attribute p|m and p|n => p|r*m and p|r*n. But r*p is in (gcd(m,n)) as well. But does that help me? I am lost.

Hypertrooper, I still don't have an answer to your problem. Have you by any chance solved it already?

I was wondering, without resorting to the definition of an isomorphism, if it is possible to (1) guess the inverse of Phi and show that it is indeed the left inverse and the right inverse of Phi, or (2) use theorems/sufficient conditions for a morphism to be an isomorphism.

I also find it interesting that the definition of Phi involves the element 1 + , and not something else. It may not help, but I was thinking of writing 1 as m * n / (gcd(m, n) * lcm(m, n)) or maybe the reciprocal of this.

 

[Pegosaurus]

hey gaf, i need some help with this physics question

1) An 8.9-kg box that has a uniform density and is twice as tall as it is wide rests on the floor of a truck. What is the maximum coefficient of static friction between the box and floor so that the box will slide toward the rear of the truck rather than tip when the truck accelerates forward on a level road?

for this the answer is Us = 0.5, but I can't figure out the second part

2) Repeat the problem if the truck accelerates up a hill that makes an angle of 9.5° with the horizontal.
&#956; s <

i can't figure out the static coefficient

i'm not good at math :x

 

[ibyea]

hey gaf, i need some help with this physics question

1) An 8.9-kg box that has a uniform density and is twice as tall as it is wide rests on the floor of a truck. What is the maximum coefficient of static friction between the box and floor so that the box will slide toward the rear of the truck rather than tip when the truck accelerates forward on a level road?

for this the answer is Us = 0.5, but I can't figure out the second part

2) Repeat the problem if the truck accelerates up a hill that makes an angle of 9.5° with the horizontal.
&#956; s <

i can't figure out the static coefficient

i'm not good at math :x

If it's not too late, can I see what you have done? I can't help you much if I don't know what part you are stuck on.

 

[Skittles]

The quiz is actually fair and does test you guys on vector spaces in a good way. I think your professor wants you to understand that you can view polynomials as vectors (how?)

Since it's a quiz, I'll give a general guide.

1a. The image of a function is how the function acts upon an input.

2a. Given a vector space V, we say that U, a subset of V, is a subspace if it is a vector space by its own. How do we show that a space is a vector space? Is there a better way to show this if we know that U is a subset of V, which is already a vector space?

3. Recall that we can view polynomials as vectors.

Am i doing this right?

 

[kgtrep]

Am i doing this right?

Skittles, your 1a is correct.

----

For 1b, if you wrote T([a b; c d]) = [0 0; 0 0] in the first line, please cross out the zero matrix, and write instead that you're trying to solve the equation,

T([a b; c d]) = (a + b) + (2d)x + (a + b)x^2 = 0 (the zero polynomial),

for a, b, c, and d.

The basis that you wrote down and the steps that you took to get there are correct.

----

You didn't write down the basis for 1c, but I think we can come up with an easier argument for the image.

----

You have the right idea for problem 3; it would be great if you explain why you are setting up a matrix (equation).

Make sure to actually show the row reduction instead of just claiming that it reduces to the identity matrix (what does getting the identity matrix mean?).

 

[Skittles]

Skittles, your 1a is correct.

----

For 1b, if you wrote T([a b; c d]) = [0 0; 0 0] in the first line, please cross out the zero matrix, and write instead that you're trying to solve the equation,

T([a b; c d]) = (a + b) + (2d)x + (a + b)x^2 = 0 (the zero polynomial),

for a, b, c, and d.

The basis that you wrote down and the steps that you took to get there are correct.

----

You didn't write down the basis for 1c, but I think we can come up with an easier argument for the image.

----

You have the right idea for problem 3; it would be great if you explain why you are setting up a matrix (equation).

Make sure to actually show the row reduction instead of just claiming that it reduces to the identity matrix (what does getting the identity matrix mean?).

Would the basis for 1c be 2+2x^2 or the columns [1,0,1] and [0,1,0]¿
Also feel like I'm still over complicating 2a, I'm pretty sure it's fairly simple

 

[Pegosaurus]

If it's not too late, can I see what you have done? I can't help you much if I don't know what part you are stuck on.

i don't even really know where to start

 

[kgtrep]

Would the basis for 1c be 2+2x^2 or the columns [1,0,1] and [0,1,0]¿
Also feel like I'm still over complicating 2a, I'm pretty sure it's fairly simple

The basis for a vector space isn't unique, so you can choose either 2 + 2x^2 or 1 + x^2 for one of the polynomial basis functions. I'd write the basis as polynomials instead of vectors, just for clarity. And yeah, the second basis function is x (or any nonzero multiple of this).

To show that a subset U of a vector space V is a subspace, we just need to show three things (instead of all the conditions for a vector space):

1. The zero vector of V is in the set U.
2. Addition of vectors from U is closed.
3. Scalar multiplication of a vector from U is closed.

Cheers,

 

[ibyea]

i don't even really know where to start

Say does the problem give you the mass of the truck? Anyways, it's the same problem except the x-y coordinates are rotated. So the normal force would be mg*cos(angle).

 

[Pegosaurus]

Say does the problem give you the mass of the truck? Anyways, it's the same problem except the x-y coordinates are rotated. So the normal force would be mg*cos(angle).

they don't give you he mass of the truck, but i managed to figure it out, thanks

 

[Skittles]

The basis for a vector space isn't unique, so you can choose either 2 + 2x^2 or 1 + x^2 for one of the polynomial basis functions. I'd write the basis as polynomials instead of vectors, just for clarity. And yeah, the second basis function is x (or any nonzero multiple of this).

To show that a subset U of a vector space V is a subspace, we just need to show three things (instead of all the conditions for a vector space):

1. The zero vector of V is in the set U.
2. Addition of vectors from U is closed.
3. Scalar multiplication of a vector from U is closed.

Cheers,

Thank you, do you know of any really good linear algebra videos I can use to brush up on my skills before finals? I know there is gilbert strang but his stuff is mostly theory with no examples.

 

[0xCA2]

I don't understand these two questions. It seems like there's so many different ways you can do this, I don't know exactly how they want it:

Note that the first box has "= 0" after it as well, it just got cut off.

My answer for the first one was was r^2-7rcos(&#415-1.

My answer for the second one would be r^2-cos(&#415-sin(&#415 (haven't tried it yet because it's mean I have to submit an answer for the previous question as well).

What am I doing wrong? How am I supposed to approach this?

 

[kgtrep]

I don't understand these two questions. It seems like there's so many different ways you can do this, I don't know exactly how they want it:



Note that the first box has "= 0" after it as well, it just got cut off.

My answer for the first one was was r^2-7rcos(&#415-1.

My answer for the second one would be r^2-cos(&#415-sin(&#415 (haven't tried it yet because it's mean I have to submit an answer for the previous question as well).

What am I doing wrong? How am I supposed to approach this?

The Cartesian coordinates (x, y) are related to the polar coordinates (r, theta) as follows:

x = r * cos(theta)
y = r * sin(theta).

Your answer to the first problem is correct, and may have just used a wrong syntax that led to a parsing error; you will likely want to enter 7 * r * cos(theta) with the multiplication symbol between terms.

Your answer to the second problem is incorrect, however. Using the relations above, please try this again.

 

[0xCA2]

The Cartesian coordinates (x, y) are related to the polar coordinates (r, theta) as follows:

x = r * cos(theta)
y = r * sin(theta).

Your answer to the first problem is correct, and may have just used a wrong syntax that led to a parsing error; you will likely want to enter 7 * r * cos(theta) with the multiplication symbol between terms.

Your answer to the second problem is incorrect, however. Using the relations above, please try this again.

For the first one, I accidentally wrote sin instead of cos. For the second one, the answer was r^2-cos(2theta) (double angle identity).

Thanks for your help!

 

[kgtrep]

For the first one, I accidentally wrote sin instead of cos. For the second one, the answer was r^2-cos(2theta) (double angle identity).

Thanks for your help!

No problem. For the second one, I'd say that r^2 * (r^2 - cos(2*theta)) = 0 is really the answer, i.e. we allow r = 0 or (x, y) = (0, 0) to be a possible solution as well.

 

[Chance Hale]

What's the best all around calculator you can get? Starting Calculus in a week and doing Physics/Calc 2 next semester and planning on doing many more math courses after that so I need to get a new one.

Is the HP Prime any good? Saw some worries about it not being a TI calculator which would make asking for help in the classroom difficult. Don't mind spending over a hundred given my last graphing calculator lasted nearly a decade. Just want something that seems relatively modern in comparison, so color would be nice

 

[Stumpokapow]

You probably don't need "the best calculator you can get" (which is just a computer and Matlab or Wolfram Alpha)--just get whatever standard TI graphing calculator they're recommending now and make sure you get one you are permitted to use on an exam because in many cases universities do not allow you to use graphing calculators which can do indefinite integration.

 

[Chance Hale]

Meant best all around more in the sense of versatility in subjects rather than the Titan X of calculators or what have you. Will probably take up to Differential Equations before I transfer into a four year program next Summer(could transfer in the Spring, but I'd rather guarantee my GPA stays in the 3.7 range by getting all the basic math classes done at the community level)

Will email the teachers I have this summer for their calculator test standards.

 

[0xCA2]

My Calculus classes have not required or allowed a calculator (all exact answers instead of approximates), so you may not need one. I spent good money on a Ti-89 a couple of years ago and it hasn't gotten much use.

Plus I find online calculators to be way more convenient and usable. That's probably the best all around calculator.

And yeah, taking your Calc classes at CC is a good idea. I wish I did

 

[loaf of bread]

convert rectangular to polar (x^2 + y^2)^2 = a(x^2)y

(x^2 + y^2)^2 = a(x^2)y
r^4 = a(x^2)y
r^4 = a (r^2)((cos(theta))^2)rsin(theta)
r = a((cos(theta))^2)sin(theta)

Is that it? Am I supposed to go further? Or is that done?

 

[kgtrep]

convert rectangular to polar (x^2 + y^2)^2 = a(x^2)y

(x^2 + y^2)^2 = a(x^2)y
r^4 = a(x^2)y
r^4 = a (r^2)((cos(theta))^2)rsin(theta)
r = a((cos(theta))^2)sin(theta)

Is that it? Am I supposed to go further? Or is that done?

Yes, but I'd stop at the third equation, and leave it as,

r^4 = a * r^3 * cos(theta)^2 * sin(theta).

As I've noted for OxCA2's problem, this allows the solution r = 0 (here, theta can take on any value), or the solution (x, y) = (0, 0) in the original equation. Your third equation implies the fourth if we assume that r does not equal to 0.

 

[themagicalkitsune]

I'm not sure how to phrase this but the following temperature problem is somehow linked to complex analysis.

Let a lamina be modeled by the upper half plane.

Find the steady state temperature T(x,y) if T has boundary values T(x, 0) = {T0, -1<x<1, T1 otherwise}

Find the heat flux into the upper half plane required to maintain the state. (or state some expression from where you can find it) Assume heat flows only along the boundary and J(x,y) = -k*Grad(T(x,y)), where k is the thermal conductivity of the lamina.

I can do the first part by applying a mobius transformation and essentially solving the problem on a separate complex plane, then transforming back to get a solution in terms of (x,y).

However my knowledge of physics is meager so I'm not sure how to start the 2nd question. All I know is divJ = 0 so that means T satisfies the Laplace equation on 2 dimensions.

 

[kgtrep]

I'm not sure how to phrase this but the following temperature problem is somehow linked to complex analysis.

Let a lamina be modeled by the upper half plane.

Find the steady state temperature T(x,y) if T has boundary values T(x, 0) = {T0, -1<x<1, T1 otherwise}

Find the heat flux into the upper half plane required to maintain the state. (or state some expression from where you can find it) Assume heat flows only along the boundary and J(x,y) = -k*Grad(T(x,y)), where k is the thermal conductivity of the lamina.

I can do the first part by applying a mobius transformation and essentially solving the problem on a separate complex plane, then transforming back to get a solution in terms of (x,y).

However my knowledge of physics is meager so I'm not sure how to start the 2nd question. All I know is divJ = 0 so that means T satisfies the Laplace equation on 2 dimensions.

Fourier's law assumes that the heat flux q = q(x, t) = q(x) is linearly proportional to the temperature gradient, grad(T) = grad(T(x, t)) = grad(T(x)):

q(x) = -k * grad(T(x)), for all x in the half-plane.

Note, I used q instead, as that's the usual notation for heat flux, and used x to denote the collection of coordinates (x, y). The minus sign reflects the physical fact that heat flows from a hot to cold area.


Hence, with the steady-state solution T(x) that you found for the first part, we can find the heat flux q(x) on the boundary y = 0 using Fourier's law.

One way to think about the result is, if we consider another problem with the same geometry (a half-plane) and same material parameters (the thermal conductivity), but, this time, prescribe this heat flux q on the boundary y = 0, then we will get the same steady-state solution as before.

(If I'm not mistaken, we should also have a remote condition, i.e. specify what happens to the temperature at infinity.)


There are many problems in mechanics whose governing equation is similar, so it may interest you to pick one and study it in detail so that you can easily make analogies to others.

Note that an infinite domain is an idealization of a "large" but finite-sized structure (large relative to some length scale). The analytical solution to the infinite-domain problem helps us approximate and understand the solution to the finite-domain problem.

 

[triplestation]

i feel like i messed this up because the final answer looks funny even if its a trig funtion *paranoid* , help is appreciated a lot

 

[Stumpokapow]

i feel like i messed this up because the final answer looks funny even if its a trig funtion *paranoid* , help is appreciated a lot

It's early and I'm on a bus so I really can't work through anything to check your final calculations, but you appear to be correct. The big thing with u substitution and definite integration are that you re-map the x-space to the u-space for the integration bounds, which you did.