LISTA DE EQUACOES UTEIS

x = cos y = sen = cos + sen j

x = r sen cos y = r sen sen z = r cos

= r

r+

r

+

r sen

=

+

+ k

z

r = sen cos + sen sen j+ cos k

~V =1

(V) +

1

V

+Vz

z

~V =1

r2

r(r2Vr) +

1

r sen

(sen V) +

1

r sen

V

~V =(1

Vz

V

z

)+

(Vz

Vz

) +

(

(V)V

) k

~V =[

(sen V)V

] rr sen

+[ 1sen

Vr

(rV)

r

] r+[ r

(rV)Vr

] r

2f =

2f

x2+2f

y2+2f

z2

2f =

1

(f

)+

1

22f

2+2f

z2

2f =

1

r

2

r2(rf) +

1

r2 sen

(sen

f

)+

1

r2 sen2

2f

2

d~ = dx + dy j+ dz k d~ = d + d + dz k d~ = dr r + r d + r sen d

dv = dxdydz dv = dddz dv = r2 sen drdd

dAz = dxdy dAy = dxdz dAx = dydz

dAz = dd dA = ddz dA = ddz

dA = r drd dA = r sen drd dAr = r2 sen dd

C

M dx+N dy =

S

(Nx

M

y

)dS A =

1

2

C

x dy y dx

V

~V dv =

S

~V n dS

V

~V dv =

S

n ~V dS

V

f dv =

S

fn dS

S

~V n dS =

C

~V d~

S

(n) ~V dS =

C

d~ ~V

S

nf dS =

C

f d~

1

( + ) = + (1a)

( ~V + ~U ) = ~V + ~U (1b)

( ~V + ~U ) = ~V + ~U (1c)

( ~V ) = ~V + ~V (1d)

( ~V ) = ~V + ~V (1e)

( ~V ~U ) = ~U ( ~V ) ~V ( ~U ) (1f)

( ~V ~U ) = ( ~U ) ~V ~U ( ~V ) ( ~V ) ~U + ~V ( ~U ) (1g)

( ~V ~U ) = ( ~U ) ~V + ~U ( ~V ) + ( ~V ) ~U + ~V ( ~U ) (1h)

= 0 (1i)

( ~V ) = 0 (1j)

2 = (1k)

( ~V ) = ( ~V )2 ~V (1l)

2