This preview shows page 1. Sign up to view the full content.
Unformatted text preview: vector . This gives us the definition of the directional is in the direction of the x-axis we just get the partial derivative with respect to x.
is in the direction of the y-axis we just get the partial derivative with respect to y. We will now derive another definition of the directional derivative that will be easier to calculate than the limit
Derivation of the Dot Product Definition for the Directional Derivative
Theory wise, the directional derivative is the slope of the tangent line to a curve, at the point
intersection of a surface
and a plane that is aligned with a unit vector ,that is the
and the unit vector .
This plane is parallel to the z-axis. Therefore the projection of the curve on the xy-plane would be the points lying along
(representing the domain of our curve)
The z-coordinates for the curve are on the surface which we can find by . Thus the composite function
will give us all points lying on the curve. Ie: this is the equation of the curve
where our parameter t can be thought of as the t-axis that is given by: Notice the curve is given by the parametric equations: We are interested in the rate of change of z as we move along the t-axis:
Thus 12.7: Tangent Planes and Linear Approximations (and what it means for a function to be Differentiable)
Finding the tangent plane to a surface at the point . To find the equation of a plane we need to know a point in the plane and a normal vector to the plane.
All tangent lines at
What does would lie in the tangent plane.
represent? This is the slope of the tangent line to the trace of in the plane . (We looked at this in section 12.3)
We can think about the partial:
change of as a fraction A change of 1 unit in the x-direction creates a units in the z-direction. This slope can be represented by the direction vector: Duplicate the logic for the other partial: Both direction vectors are parallel to the tangent plane. So we can use them to find a vector that is perpendicular to the
tangent plane. Remember the cross product of two vectors creates a third vector th...
View Full Document
- Spring '08