I got a problem that I have to determine the tangent plane, normal line and gradient of a vector function in a specific point.
This problem gives me this vector function $h$:
$h: (0, +\infty) \times (0, +\infty) \to \mathbb{R}^2$
$h(x,y)=(x^y,x^x\sinh(y))$
and the point $(1, 1)$.
I don't even know how to start solving this problem.
Best Answer
The graph of the function is the surface with parametrization
$$H: (0,+\infty)^2 \rightarrow\mathbb{R}^4$$
given by $H(x,y) = (x, y, x^y, x^x\sinh(y))$. The tangent plane at each point of the surface is generated by the partial derivatives of $H$, which are
$\frac{\partial H}{\partial x} = (1, 0, yx^{y-1},x^x(1+\log x)\sinh(y))$
$\frac{\partial H}{\partial y} = (0,1, x^y\log x,x^x\cosh(y))$
At the point $(1,1)$, they are the tangent vectors
$\frac{\partial H}{\partial x}(1,1) = (1,0,1,\sinh(1))$
$\frac{\partial H}{\partial y}(1,1) = (0,1,0,\cosh(1))$
So, points on the tangent plane at that point are of the form $(1,1) + t(1,0,1,\sinh(1)) + s(0,1,0,\cosh(1))$, for arbitrary $s, t$.
The normal line still makes no sense, because the orthogonal space of a plane in $\mathbb{R}^4$ has $2$ dimensions and not one (it would be a "normal plane"). Same thing for the "gradient". You could calculate the gradients of the components, which would be the partial derivatives of each component, and I already got them:
$grad H_1 = (yx^{y-1}, x^y\log x)$
$grad H_2 = (x^x(1+ \log x)\sinh(y), x^x\cosh(y))$