Homework Statement
show that
\int^{∞}_{0}\frac{sin^{2}x}{x^{2}}dx= \frac{\pi}{2}
Homework Equations
consider
\oint_{C}\frac{1-e^{i2z}}{z^{2}}dz
where C is a semi circle of radius R, about 0,0 with an indent (another semi circle) excluding 0,0.
The Attempt at a Solution...
F = \nabla\times A = \nabla\times1/2F x r
1/2F x r = 1/2εijkFjrk
F= \nabla\times 1/2εijkFjrk
= εimn∂/∂xi(1/2εijkFjrk)m
=∂/∂xi(δmjδnk-δnjδmk)(1/2Fjrk)m
=(1/2)∂/∂xi(Fmrn - Fnrm)m
=?
This is another problem I'm a bit stuck on with similar content
Homework Statement
F is a constant vector field. hence \nabla\cdot F = 0
this means there is a vector potential F = \nabla\times A
also \nabla\times F = 0
this means there is a scalar potential F = \nabla ∅
verify
∅ = F...
well I'll assume thats close to right and move onto the next problem which seems to make sense
Homework Statement
\nabla\cdot \vec{r}= δii=3
Homework Equations
∂xi/∂xj=δij
The Attempt at a Solution
\nabla\cdot \vec{r}=\nabla_{i}r_{i}
= ∂/∂xiri
= ∂xi/∂xi = δii...
Still struggling with the concept. So I can use a kronecker delta to have the condition of index l being j...
\frac{∂}{∂x}(δ_{jl}x_{l}x_{l})_{j} =
( δ_{jl}=1 if l=j else 0)
\frac{∂}{∂x}x_{j}x_{j} =
2x_{j}