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1. ### Miscellaneous Definite Integrals

I mean, differentiating the top and bottom before taking the limit. as in a taylor series?
2. ### Miscellaneous Definite Integrals

The teacher gave me a hint, use L'hopitals Rule, which isn't clear to be possible since i forgot to add in the limit
3. ### Miscellaneous Definite Integrals

Thank you. I was half awake when I wrote that. I just don't know how to get that ε out from the denominator, I cannot take the limit otherwise.
4. ### Miscellaneous Definite Integrals

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...
5. ### Proving vector calculus identities using summation notation

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 =?
6. ### Proving vector calculus identities using summation notation

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...
7. ### Proving vector calculus identities using summation notation

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...
8. ### Proving vector calculus identities using summation notation

That makes sense! So you do it that way because you can't have x2 instead you do product rule = 2(∂ xl/∂xj) xl = δjlxl if l=j = 2xj
9. ### Proving vector calculus identities using summation notation

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}
10. ### Proving vector calculus identities using summation notation

hmmm so \frac{∂}{∂x}(x_{l}x_{l})_{j} = \frac{∂}{∂x}(x_{l}^{2})_{j} = \frac{∂}{∂x}x^{2}_{l}δ_{jl} = if l=j δ_{jl}=1 else δ_{jl}=0 \frac{∂}{∂x}x^{2}_{j} = 2x_{j}
11. ### Proving vector calculus identities using summation notation

Ok, sounds good. so 3. Attempt at solving (\nablar^{2})_{j}= \frac{∂r^{2}}{∂x_{j}}= \frac{∂}{∂x_{j}}(δ_{lm}x_{l}x_{m})= \frac{∂}{∂x_{j}}(x_{l}x_{l}) = \frac{∂}{∂x_{j}}(x^{2}) = 2x_{j}
12. ### Proving vector calculus identities using summation notation

Homework Statement \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Equations \vec{r} = x_{i}e_{i} The Attempt at a Solution \frac{∂x_{i}}{∂x_{j}} = 1 iff i=j δ_{ij} = 1 iff i=j therefore \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Statement r^{2} = x_{k}x_{k} Homework...