find tangent vector HELP!!!?

Question

find the 2 tangent vectors to:
X = (cosh(A)cos(B) , cosh(A)sin(B) , sinh(A)) at A=B=1
hence find the tangent plane at this point

Answer 1

What you want are the two partial derivatives: of X with respect to A and of X w.r.t. B.

When you take the partial derivative of X w.r.t A, terms in B are taken as constant, while when taking the derivative w.r.t. B, the terms in A are constant.

Thus:
dX/dA(cosh(A)cos(B)) = (cos(B))(dX/dA(cosh(A)) = (cos(B))(sinh(A))
dX/dB(cosh(A)cos(B)) = (cosh(A))(dX/dB(cos(B))) = (cosh(A))(sin(B))

When you are taking the (partial) derivative of a vector valued function, take the derivative of each component separately:

F(A,B) =

dF/dA =

So the two vectors you want are: and , where Fx(A,B) = cosh(A)cos(B), etc.