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About LinkBacksHow do you show that the tangent plane at the point,
on the surface z=f(x,y) defined by a differentiable function f is the plane:
OR
+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0}))
Let's look at the partial derivatives and see what we can understand from them. Given some function,
measures the rate of change of
with respect to
, and
measures the rate of change of
. Now we will look for vector-valued functions of tangent lines at the point
for
for
. Let's just choose
. We could pick any constant, but let's choose 1 for ease. This is just saying that the x-coordinate of the tangent line to the curve created by the
since y will never change.
will be the slope of the tangent at the point
. Let's call this vector
. Following the logic of the
. These vectors are not parallel and they are not skew (they intersect at the point
. So they must lie on some plane that will be tangental to the point. To get the equation of the plane, we need the point and the normal vector. The normal vector is simply the cross product of those two tangent vectors.
. So we have our point
We can rewrite this as:
+\frac{\partial f}{\partial x}(a,b)(x-a)+\frac{\partial f}{\partial y}(a,b)(y-b))
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