If A=(x^4 - y^2*z^2) i + (x^2+ y^2) j - (x^3*y^3*z^3 ) k,  determine curl of A at the point (1,4,-3).

Expert Answers
kjcdb8er eNotes educator| Certified Educator

The mathematical curl of a vector function A = Ax i + Ay j + Az k is defined as:

curl A = (dAz/dy - dAy/dz)i + (dAx/dz - dAz/dx)j + (dAy/dx - dAx/dy)k

Calculate the above derivatives and plug in for the point (1,4,-3)

dAz/dy = -3x^3y^2z^3 =  -3(4)^2(-3)^3 = 1296

dAy/dz = 0

dAx/dz = -2y^2z = -2(4)^2(-3) = 96

dAz/dx = -3x^2y^3z^3 = -3(1)^2(4)^3(-3)^3 = 5184

dAy/dx = 2x = 2(1) = 2

dAx/dy = -2yz^2 = -2(4)(-3)^2 = -72

Curl A = (1296 - 0 )i + (96 - 5184)j + (2 + 72)k

Curl A = 1296i - 5088j + 74k

neela | Student

Curl  of  vector A = [ (i     j     k),  (db/dbx , db/dy ,  db/dz), ( A1i  , A2j,  A3k)]

 Where  vector A  = A1i+A2j+A3k

Curl of vector A = (dbA3/dby -dbA2/dbz)i + (dbA1/dbz - dbA3/dbx)j + (dbA2/dbx-dbA1/dy)]

= i {db/dy(-x^3y^3*z^3) - db/dz(x^2+y^2)}

+ j {db/dbz(x^4-y^2z^2) - db/dbx (-x^3y^3z^3)}

k { db/dbx ( x^2+y^2) - db/dy (x^4-y^2z^2)}

= i { (-x^3*3y^2z^3) - ( 0) }

+j { (-y^2*2z) -  (-3x^2y^3z^3)}

+k{ (2x) - (2yz^2}.

=i( - 3x^3*y^2*z^3) +j(-2y^2z +3x^2*y^3*z^3)+k(2x-2yz^2)

Threfore curl of A at (1, 4, -3) = i(-3*4^2(-3)^2) + J(-2*4^2*(-3)+3*4^3*(-3)^3)+k(2-2*3(-3)^2)

= i(-432) +j(96 - 5184)+k(2-54)

= (-432)i + (-5088)j + (-52) k is the Curl of A at (1,4,-3).