Which of the following represents a Pythagorean Triple? 1 {10, 20, 30} {5, 7, 10} {21, 72, 75} {11, 45, 60}

Answer:

F=dA  

dF=0  

d⋆F=⋆J  

where in fact the second equality is not needed because of the identity  d2=0 . In abstract tensor notation, these equations expand to

Fab=2∇[aAb]  

∇[aFbc]=0  

∇aFab=Jb  

where in the above, additional compact notation has been used to simplify things: square brackets mean antisymmetrizing indices, and the covariant derivative  ∇a  is related to a flat derivative  ∂a  and a connection  Γbac . Now, performing a 3+1 decomposition (here in flat space-time) Maxwell's equations break down as

∇⋅B⃗ =0  

∇⋅E⃗ =4πρ  

∂B⃗ ∂t=−∇×B⃗  

∂E⃗ ∂t=+∇×B⃗ −4πJ⃗ .  

In the above, there is yet more compactifying notation: the divergence is defined as

∇⋅v⃗ =∂vi∂xi  

and the curl is defined as

(∇×v⃗ )i=ϵijk∂vk∂xj.  

In the above, still more compactifying notation has been used, namely the Einstein summation convention. Written out in components, the divergence is

∇⋅v⃗ =∂vx∂x+∂vy∂y+∂vz∂z  

and curl is defined as

(∇×v⃗ )z=∂vy∂x−∂vx∂y  

and similarly for the other two components.

As you can see, the extremely compact and powerful notation

dF=0  

d⋆F=⋆J  

expands out to a large set of component equations—but they have the same content, so it's not very meaningful that the equations in components are much longer than in terms of differential forms.

As another example, Elson Liu answered with an exploded-out form of the Standard Model Lagrangian. However, it can be written much more compactly:

L=LYM+LHiggs+Llep.+Lqrk.+LYuk.  

where

LYM=−14TrFabFab  

LHiggs=−(Daϕ†)(Daϕ)+14λ(ϕ†ϕ−12v2)2  

Llep.=iℓ†I/DℓI+ie¯†I/De¯I  

Lqrk.=iq†I/DqI+iu¯†I/Du¯+id¯†I/Dd¯I  

LYuk.=−yIJϕℓIe¯J−y′IJϕqId¯J  

−y′′IJϕ†qIu¯J+h.c.  

and specifying that F is the curvature on the gauge group  G≃SU(3)⊗SU(2)⊗U(1) ,  Da  is the gauge covariant derivative on G,  /D  is the gauge covariant Dirac operator on G, and specifying the representations of the fields: the scalar  ϕ  in the rep (1,2,-1/2), and the fermions  ℓ  in the rep (1,2,-1/2),  e¯  in (1,1,+1),  q  in (3,2,+1/6),  u¯  in  (3¯,1,−2/3) , and  d¯  in  (3¯,1,+1/3) ; where I is a generation index running over 1,2,3, and where everything else is a constant.

Step-by-step explanation:

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City A and City B had two different temperatures on a particular day. On that day, four times the temperature of City A was 8° C more than three times the temperature of City B. The temperature of City A minus twice the temperature of City B was −3° C. The following system of equations models this scenario:

4x = 8 + 3y
x − 2y = −3

What was the temperature of City A and City B on that day?

City A was 5° C, and City B was 4° C.
City A was 3° C, and City B was −1° C.
City A was 8° C, and City B was −3° C.
City A was 5° C, and City B was −5° C.

(7q+7)+(9q+1)
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Darin is tired of his CDs getting scratched up and not playing properly. He wants to purchase an mp3 player to solve this problem. The one he wants costs $400. Darin has $175.25 in his checking account. He plans on depositing $40.50 into his checking account each week. Let w = number of weeks. Determine if the statement is true by selecting Yes or No.
answers below:
A) The inequality 175.25 + 40.50w ≥ 400 can
be used to find the number of weeks until
Darin can purchase the mp3 player.
Yes
No
B) The inequality 175.25 + 40.50w > 400 can
be used to find the number of weeks until
Darin can purchase the mp3 player.
Yes
No
C) The inequality 175.25 + 40.50w ≤ 400 can
be used to find the number of weeks until
Darin can purchase the mp3 player.
Yes
No
D) The inequality 175.25 + 40.50w < 400 can
be used to find the number of weeks until
Darin can purchase the mp3 player.
Yes
No
E) It will take Darin 5 weeks until he can
purchase the mp3 player.
Yes
No
F) It will take Darin 6 weeks until he can
purchase the mp3 player.
Yes
No

Solve the inequality.

2x + 7 ≥ -13


A. x ≥ -6
B. x ≤ -10
C. x ≤ -6
D. x ≥ -10

Mr. O’Reilly has a cup of coffee at a diner. The cup of coffee costs $1.25, and he leaves a 60% tip. How much does he leave for a tip?

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solve for x: 3 x - s = r

Miguel bought a table on sale for $266. This price was 65% less than the original price.
What was the original price?

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Use the key features of the polynomial f(x) = 6x4 + 4x3 − 5x2 − 2x + 1 to describe its end behavior.

The left side continues up, and the right side continues up.
The left side continues down, and the right side continues down.
The left side continues up, and the right side continues down.
The left side continues down, and the right side continues up.

Mallorie is measuring the rug in her bedroom with a yardstick. She gets 2 yards the first time she measures, then she gets 6 feet the second time she measures. Qhy are the lengths different and no picture included

I NEED HELP FAST! What is the circumference of the circle below?

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What is the sales tax for a purchase of $2,500?

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FIND THE AREA OF THE SHAPE AND THE PART THAT DOES NOT HAVE the empty part of the circle that says 6 on it

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Find the value of y (view picture below)

How much interest will Jesse pay if he borrows $5,000 for 5 years with an interest rate of 8%?

A. $2,000.00
B. $20,000.00
C. $200.00
D. $2,500.00

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