What Are the Differences Between Black Body Radiation Formulas?

[sss1]

Homework Statement

Below

Relevant Equations

In the pictures

Im getting confused between the differences of all of these formulas.
I googled spectral radiance black body and all of the first four pictures came up. They represent the intensity of radiation at a particular wavelength right, or the y-axis of the black body radiation curve? So if I integrate this formula I should get the total intensity? Or the total area under the black body radiation curve? One of the pictures has frequency as the variable instead of wavelength tho? Is it finding the same thing but for when I'm given frequency instead of wavelength? And somehow the rest of the three pictures all have different numerators...?
And the last formula, which finds the radiated power for a specific wavelength, why does it have a delta lambda in it? Kinda confused on where it comes about. I understand that spectral radiancy has units Watts/m^3, so it makes sense to have A and delta lambda, because that has units m^3. But why not have lambda instead of delta lambda? And also if i integrated that formula it will give me Stefan Boltzmann's law? The total power radiated?

 

[haruspex]

Homework Statement: Below
Relevant Equations: In the pictures

Im getting confused between the differences of all of these formulas.
I googled spectral radiance black body and all of the first four pictures came up. They represent the intensity of radiation at a particular wavelength right, or the y-axis of the black body radiation curve? So if I integrate this formula I should get the total intensity? Or the total area under the black body radiation curve? One of the pictures has frequency as the variable instead of wavelength tho? Is it finding the same thing but for when I'm given frequency instead of wavelength? And somehow the rest of the three pictures all have different numerators...?
And the last formula, which finds the radiated power for a specific wavelength, why does it have a delta lambda in it? Kinda confused on where it comes about. I understand that spectral radiancy has units Watts/m^3, so it makes sense to have A and delta lambda, because that has units m^3. But why not have lambda instead of delta lambda? And also if i integrated that formula it will give me Stefan Boltzmann's law? The total power radiated?

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Unfortunately, the way you have posted those images they cannot be clicked to reveal the whole text. If you cannot figure out how to do it, post the links.

 

[sss1]

Unfortunately, the way you have posted those images they cannot be clicked to reveal the whole text. If you cannot figure out how to do it, post the links.

Does it work now?

 

[haruspex]

Does the table at https://en.wikipedia.org/wiki/Planck's_law#Different_forms help?

 

[sss1]

Does the table at https://en.wikipedia.org/wiki/Planck's_law#Different_forms help?

I had a look at the table and tried calculating the spectral radiancy using both frequency and wavelength, but got different answers?
I used these two formulas.
The wavelength I used was 966e-9m, and so the frequency should be (3e8)/(966e-9) Hz?

 

[haruspex]

I had a look at the table and tried calculating the spectral radiancy using both frequency and wavelength, but got different answers?
I used these two formulas.
The wavelength I used was 966e-9m, and so the frequency should be (3e8)/(966e-9) Hz?

$B_\nu$ is the spectral emissive power per unit area, per unit solid angle and per unit frequency. $B_\lambda$ is per unit wavelength.
I.e. $B_\nu d\nu$ is the total spectral emissive power per unit area, per unit solid angle for the frequency range $(\nu,\nu+d\nu)$, etc.
Hence $B_\nu =B_\lambda|\frac{d\lambda}{d\nu}|=B_\lambda\frac{c}{\nu^2}$.
If you make that substitution, and $\nu=\frac c{\lambda}$, you should see one equation turn into the other.