\magnification=\magstep1

$$
E_\lambda \ d\lambda \quad = \quad
  I_\lambda \thinspace d \lambda \thinspace dt \thinspace 
             dA \cos(\theta) \thinspace d\Omega
$$
              
$$
<I_\lambda> \quad = \quad 
                                 {{1}\over{4 \pi}}  \   
                           \int I_\lambda \thinspace d\Omega  
  \quad = \quad 
                                 {{1}\over{4 \pi}}  \   
                           \int_0^{2 \pi} d \phi \  
                           \int_0^{\pi}  I_\lambda \sin(\theta) d \theta 
$$

$$
<I_\lambda> \quad = \quad 
                                 {{1}\over{4 \pi}}  \  I_\lambda \   
                           \int_0^{2 \pi} d \phi \  
                           \int_0^{\pi}  \sin(\theta) d \theta 
$$

$$
u_\lambda \ d \lambda \quad = \quad
                         {{4 \pi}\over{c}} \ <I_\lambda> \thinspace d\lambda
$$

$$
<I_\lambda> d\lambda \quad {\rm has\ units\ }\quad\quad 
                  {{\rm J}\over{\rm s \ m^2}}
$$

$$
{{4 \pi}\over{c}} \ <I_\lambda> d\lambda \quad {\rm has\ units\ }\quad\quad 
                  {{1}\over{\rm m/s}} \ {{\rm J}\over{\rm s \ m^2}}
    \quad = \quad 
                  {{\rm s}\over{\rm m}} \ {{\rm J}\over{\rm s \ m^2}}
    \quad = \quad 
                  {{\rm J}\over{\rm m^3}}
$$


$$
u_\lambda \ d \lambda \quad = \quad
                    \int_{4000 \AA}^{8000 \AA} 
                         {{4 \pi}\over{c}} \ B_\lambda (T) \thinspace d\lambda
$$


$$
u_\lambda \ d \lambda \quad = \quad
                    \int_{5 {\rm \mu m}}^{20 {\rm \mu m}} 
                         {{4 \pi}\over{c}} \ B_\lambda (T) \thinspace d\lambda
$$

$$
u_\lambda \ d \lambda \quad = \quad
                    \int_{0}^{\infty} 
                         {{4 \pi}\over{c}} \ B_\lambda (T) \thinspace d\lambda
           \quad = \quad
                       {{4 \sigma}\over{c}} \ T^4
           \quad = \quad
                       7.566 \times 10^{-16} \ {\rm J \ m^{-3}} \ T^4
$$


$$
F_\lambda d\lambda \quad = \quad \int I_\lambda d\lambda \thinspace
               \cos(\theta) \thinspace d \Omega
$$

              
              
              
              


\bye
