atm1993 said:Homework Statement
Find an equation of the tangent plane to r(u,v) = uvi + ue^vj + ve^uk at the origin, (0,0,0).
Homework Equations
The Attempt at a Solution
I don't really know how to start. I've been reading and searching around for quite a bit. I know how to do the problem with a regular function, using the partial derivatives of a function and such, but I'm not really sure where to go with this vector-valued function.
atm1993 said:Alright, so I worked it out and I want to see if I'm on the right track following what you said.
For the function to = <0,0,0>, (u,v) would have to be (0,0). I then found the partials of the function with respect to u and v.
ru(u,v) = vi + e^vj + ve^uk
and
rv(u,v) = ui + ue^vj + e^uk
I then evaluated both of those at (0,0), resulting in
<0, 1, 0> and <0, 0, 1>.
Crossing those resulted in <1, 0, 0>, which then leaves me with an equation of a plane as x=0, using the point (0, 0, 0).
I'm not completely sure if I did that correctly.
The equation of the tangent plane for a vector-valued function is given by z - z0 = ∇f(x0, y0).(x-x0, y-y0) , where z0 is the value of the function at the given point and ∇f(x0, y0) is the gradient vector evaluated at that point.
The gradient vector for a vector-valued function is found by taking the partial derivatives of the function with respect to each variable and putting them together in a vector. For example, if the function is given by f(x,y) = x2 + 2y, then the gradient vector is ∇f(x,y) = (2x, 2) .
The equation of the tangent plane provides information about the slope or rate of change of the function at a specific point. It also gives the local approximation of the graph of the function near that point.
Yes, the normal vector to the surface can be found using the gradient vector from the equation of the tangent plane. The normal vector is given by n = ∇f(x0, y0) = (fx, fy, -1) , where fx and fy are the partial derivatives of the function with respect to x and y, respectively.
The equation of the tangent plane is used in applications such as optimization problems, where it helps to find the maximum or minimum value of a function. It is also used in physics and engineering to approximate the behavior of a system at a specific point. Additionally, it is used in computer graphics to create 3D models of surfaces.