Understanding Vector Equations in Homework Problems

[quietrain]

Homework Statement

Homework Equations

The Attempt at a Solution

can anyone enlighten me what the red and blue equations mean? i remembered something about the red eqn meaning unit vector in vector format, or scalar format is it?

and the blue eqn i have totally no idea what my prof is doing >< is he trying to say that del F is a dot product and since they are 0 because i =/= j, that means that cross product is 0?

help appreciated!

 

[tiny-tim]

hi quietrain !

it's all very simple …

r^/r² = (r/|r|)/r² = r/r³

and

∂/∂x_i (F_j)

= ∂/∂x_i (x_j)

= ∂/∂x_i (x_j/r³)

= {∂/∂x_i (x_j)}/r³ + {∂/∂x_i 1/r³)}x_j (product rule)

= 0 + {∂/∂x_i 1/r³)}x_j

 

[quietrain]

with regards to part b)

for the cross product (del X F) to be 0, are they saying that del_yF_z - del_zF_y must be 0 and the other permutations too,

so that del_yF_z = del_zF_y

but from the form ∂/∂xi (Fj), it has a minus sign after differientiating, thus for Eijk, if we swop once to Eikj , then the minus sign is gone

so now its minus - plus = 2 minus , not 0?

 

[tiny-tim]

hi quietrain!

(have a del: ∇ and an epsilon: ε )

for the cross product (del X F) to be 0, are they saying that del_yF_z - del_zF_y must be 0 and the other permutations too,

so that del_yF_z = del_zF_y

yes

but from the form ∂/∂xi (Fj), it has a minus sign after differientiating, thus for Eijk, if we swop once to Eikj , then the minus sign is gone

so now its minus - plus = 2 minus , not 0?

ah, you can either write ∇_jF_k - ∇_kF_j

or ε_ijk∇_jF_k …

same thing ​

 

[quietrain]

hi quietrain!

(have a del: ∇ and an epsilon: ε )


yes


ah, you can either write ∇_jF_k - ∇_kF_j

or ε_ijk∇_jF_k …

same thing ​

oh so it becomes minus minus minus which is minus plus = 0
thanks!