Can you help me with this equality?

• Abrain
In summary: But I'm still not sure what dS is.This could be... the book does'n say exactly what dS is.However dS = -dx-dy-dz+cdt is a 4-volume, isn't it?But,if you are right, the equality is holding only because of the Gauss law? This sound pretty correctOK, apparently this really is supposed to be the charge continuity equation; the time rate change of charge contained in a volume is equal to the charge entering.If so, the left hand side needs a d/dt operator and S would be the surface normal vector having units of area pointing inward. But I'm still not sure what dS is.
Abrain
I'd like to understand why $$\int{\rho}dV = \frac{1}{c} \int{j^0}dV = \frac{1}{c} \int{j^i}dS_i$$ (the second equality), where

$$j^i = \rho \frac{dx^i}{dt}$$ is the current density 4-vector

$$\mathbf{j} = \rho \mathbf{v}$$ is the current density 3-vector

$$j^i = (c\rho, \mathbf{j})$$

$$\rho$$ is the charge density

Are you able to explain me this equality?

Thank you very much!

Last edited:
The tags need to be "tex" or "itex" instead of "latex". You also need to know that there's a bug that makes the wrong images appear in the previews most of the time. The workaround is to refresh and resend after each preview. You can edit your posts for 24 hours, so there's still time to fix the one above.

Fredrik said:
The tags need to be "tex" or "itex" instead of "latex". You also need to know that there's a bug that makes the wrong images appear in the previews most of the time. The workaround is to refresh and resend after each preview. You can edit your posts for 24 hours, so there's still time to fix the one above.

It doesn't seem to work neither with $$or [itex] I think your problem is that you've got the wrong kind of slash in the closing tags, should be /tex not \tex (and don't use capital TEX). Let's see if changing those works:I'd like to understand why [tex] \int{\rho}dV = \frac{1}{c} \int{j^0}dV = \frac{1}{c} \int{j^i}dS_i$$ (the second equality), where

$$j^i = \rho \frac{dx^i}{dt}$$ is the current density 4-vector

$$\mathbf{j} = \rho \mathbf{v}$$ is the current density 3-vector

$$j^i = (c\rho, \mathbf{j})$$

$$\rho$$ is the charge density

$$dS_i$$ is the element $$-dx-dy-dz+cdt$$

Are you able to explain me this equality?

Thank you very much!

JesseM said:
I think your problem is that you've got the wrong kind of slash in the closing tags, should be /tex not \tex (and don't use capital TEX).

Abrain said:
$$\int{\rho}dV = \frac{1}{c} \int{j^0}dV = \frac{1}{c} \int{j^i}dS_i$$

Your units don't seem to match-up. I take V to be in units of volume. In such case, S should also have units of volume.

Rather instead, I think you have mistakenly taken an element of S to be it's surface normal vector, and S should be the boundary of the volume V. Does this sound close?

Phrak said:
Your units don't seem to match-up. I take V to be in units of volume. In such case, S should also have units of volume.

Rather instead, I think you have mistakenly taken an element of S to be it's surface normal vector, and S should be the boundary of the volume V. Does this sound close?

This could be... the book does'n say exactly what dS is.
However $$dS = -dx-dy-dz+cdt$$ is a 4-volume, isn't it?

But,if you are right, the equality is holding only because of the Gauss law? This sound pretty correct

OK, apparently this really is supposed to be the charge continuity equation; the time rate change of charge contained in a volume is equal to the charge entering.

If so, the left hand side needs a d/dt operator and S would be the surface normal vector having units of area pointing inward.

1. What is an equality?

An equality is a mathematical concept that represents the idea that two expressions or values are equal to each other.

2. How do I solve an equality?

To solve an equality, you need to isolate the variable on one side of the equation by using inverse operations. This will help you find the value of the variable that makes the equation true.

3. What are the different types of equalities?

The two main types of equalities are numerical equalities, which involve numbers and arithmetic operations, and algebraic equalities, which involve variables and algebraic operations.

4. How can I check if an equality is true?

You can check if an equality is true by substituting the value of the variable into the equation and evaluating both sides. If they are equal, then the equality is true.

5. What are some common mistakes when solving equalities?

Some common mistakes when solving equalities include forgetting to perform the same operation on both sides of the equation, making a mistake in the order of operations, and forgetting to distribute a negative sign when using the distributive property.

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