Vector Calculas

Question

Let r=xi^+yj^+zk^ and r=||r||.

Show that:

∇(lnr)=r/r².

and

∇×(rⁿr)=0.

Answer 1

Let r=xi^+yj^+zk^ and r=||r||.

Show that:

∇(lnr)=r/r².

and

∇×(rⁿr)=0.

the key to first problem is to say ln(r)=integral from r=1 to r=r of (x²+y²+z²)^-(1/2)dr

for partial derivative d(ln(r))/dx ,,,

d(ln(r))/dx= [d(ln(r))/dr]*dr/dx= [(x²+y²+z²)^-(1/2)]*[2x*(1/2)* (x²+y²+z²)^-(1/2)]

so d(ln(r))/dx=x/r²

do the same for d(ln(r))/dy and d(ln(r))/dz

∇(lnr)= [d(ln(r))/dx, d(ln(r))/dy, d(ln(r))/dz] the gradient is vector,, same as ∇(lnr)=r/r²

the second problem requires a determinant be set up

first row is i,j,k

second row is r_x,r_y,r_z

third row row is x*rⁿ,y*rⁿ,z*rⁿ

r_x=x* (x²+y²+z²)^-(1/2)

r_y=y* (x²+y²+z²)^-(1/2)

r_z=z* (x²+y²+z²)^-(1/2)