Vectors and Coordinates

Question 1

What are the cylindrical coordinates?

Answer 1

x = R cos φ , y =Rsinφ , z = z
R = (x^2 + y^2)^1/2
, φ = tan^−1(y/x)

Question 2

What is the relation between R,φ and z?

Answer 2

R=φ x z
φ=z x R
z= R x φ

Question 3

What is dV for cylindrical coordinates?

Answer 3

dV=R dR dφ dz

Question 4

What is dS for cylindrical coordinates?

Answer 4

dS=R dφ dz

Question 5

What are the spherical coordinates?

Answer 5

x = r cos φ sin θ
y = r sin φ sin θ
z = r cos θ
r = (x^2 + y^2 + z^2)^1/2
φ = tan^−1(y/x) ,
θ = cos^−1(z/r)

Question 6

What is the relation between r,φ and θ?

Answer 6

θ=φ x r
φ=r x θ
r= θ x φ

Question 7

What is dV for spherical coordinates?

Answer 7

dV=r sinθ dr dφ dθ

Question 8

What is dS for spherical coordinates?

Answer 8

dS= r^2 sinθ dθ dφ

Question 9

What is the path integral for a scalar?

Answer 9

∫f(r) dℓ =∫f(s)dr/ds ds
Integrated over the path and the start and end points where f is a scalar and r is a vector

Question 10

What is the path integral for a vector?

Answer 10

∫G(r) . dr =∫G(s) . dr/ds ds
Integrated over the path and the start and end points where G is a vector and r is a vector

Question 11

What is the surface integral for a scalar?

Answer 11

∫∫f(r) dS =∫∫f(u,v) abs(dr/du x dr/dv) dv du
Integrated over the path and the start and end points where f is a scalar and r is a vector

Question 12

What is the surface integral for a vector?

Answer 12

∫∫G(r) . dS =∫∫G(u,v) . abs(dr/du x dr/dv) dv du
Integrated over the path and the start and end points where G is a vector and r is a vector