Solve 4 Vector Problem: Wave & Continuity Equations

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In summary, the continuity and wave equations can be expressed in tensor notation as shown above, and the value of j4 would depend on the specific situation.
  • #1
neelakash
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Homework Statement



I cannot remember if I have already posted this question here...
However,if that is the case,let it be said that I want to discuss more about it.

Let us define 4 vector by 4 co-ordinates (x1,x2,x3,x4) where (x1,x2,x3) are space components (like x,y,z) and x4 is related to time as x4=ict.Express the following equations in tensor notation.

(i)The continuity equation: div J+(del*rho/del t)=0

(ii)The wave equation del2ø-(1/c2) [del2 ø/del t2]=0

(iii)What will be the value of j4 instead of above?

Homework Equations



The Attempt at a Solution



For the first two part I am getting these:

Please check...

Let me know if I am correct.


(i) (∂/∂x_i)(J_i)+ (∂/∂t)ρ=0


(ii) [∂^2/(∂x_i*∂x_i)]ψ-(1/c^2)[(∂^2/∂t^2)ψ]=0

I have changed phi to psi...

Do not mind.

How can I get the 3rd part?
 
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  • #2


it is important to provide accurate and helpful responses to forum posts in order to promote discussion and learning. Therefore, I would like to provide a suggestion for expressing the continuity equation and wave equation in tensor notation, and also address the third part of the question.

For the continuity equation, it can be expressed as:

(∂J_i/∂x_i) + (∂ρ/∂t) = 0

In tensor notation, this can be written as:

∂J_i/∂x_i + ∂ρ/∂t = (∂J_i/∂x_i, ∂ρ/∂t)

Similarly, for the wave equation, it can be expressed as:

(∂^2/∂x_i∂x_i)ψ - (1/c^2)(∂^2/∂t^2)ψ = 0

In tensor notation, this can be written as:

∂^2ψ/∂x_i∂x_i - (1/c^2)∂^2ψ/∂t^2 = (∂^2ψ/∂x_i∂x_i, -1/c^2 ∂^2ψ/∂t^2)

As for the third part of the question, the value of j4 would depend on the specific situation and cannot be determined without more information. It could represent the fourth component of the current density vector in certain coordinate systems, but without more context, it is difficult to provide a definitive answer. I would suggest providing more information or clarifying the question for a more accurate response.
 

What are vector problems in relation to wave and continuity equations?

Vector problems in wave and continuity equations involve the use of vectors, which are mathematical quantities that have both magnitude and direction. These problems typically require the application of vector algebra and calculus to solve.

What is the wave equation and how is it used in vector problems?

The wave equation is a mathematical formula that describes the behavior of waves. It is used in vector problems to determine the direction and magnitude of a wave's movement at a given point in time.

How are continuity equations applied in vector problems?

Continuity equations are used in vector problems to ensure that the flow of a vector quantity, such as mass or energy, is conserved over time. This helps to determine how a vector changes in space and time.

What are some common techniques for solving vector problems in wave and continuity equations?

Some common techniques for solving vector problems in wave and continuity equations include using vector algebra, calculus, and differential equations. Additionally, numerical methods and computer simulations can also be used to solve these problems.

What real-world applications can be modeled using vector problems in wave and continuity equations?

Vector problems in wave and continuity equations have many real-world applications, including predicting the behavior of sound waves, analyzing fluid flow in pipes, and understanding the movement of electromagnetic waves. These equations are also used in fields such as engineering, physics, and meteorology.

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