First, you will hopefully recall from the Quadric Surfaces section that this is an elliptic paraboloid that opens downward. If the normal direction is denoted by
n
{\displaystyle \mathbf {n} }
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, then the normal derivative of a function f is sometimes denoted as
f
n
{\textstyle {\frac {\partial f}{\partial \mathbf {n} }}}
. For instance, one could be changing faster than the other and then there is also the issue of whether or not each is increasing or decreasing.
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Let’s take a quick look at an example. ▽v(f+g)=▽vf+▽vgThis is also known as Leibnizs rule. For our example we will say that we want the rate of change of \(f\) in the direction of \(\vec v = \left\langle {2,1} \right\rangle \). It is also a much more general formula that will encompass both of the formulas above.
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u\)Let the function \(z=f(x, y)\) is differentiable at \(\left(x_{0}, y_{0}\right)\), then; If \(\vec{\nabla} f\left(x_{0}, y_{0}\right)=\overrightarrow{0}\), then \(D_{\vec{u}} f\left(x_{0}, y_{0}\right)=0\) for any unit vector \(\vec{u}\). The gradient of a specific function at a point is go to this site vector that points in the direction along which the slope of the function is maximum. Next, we need the unit vector for the direction,Finally, the directional derivative at the point in question is,Before proceeding let’s note that the first order partial derivatives that we were looking at in the majority of the section can be thought of as special cases of the directional derivatives. The definition of the directional derivative is,So, the definition of the directional derivative is very similar to the definition of partial derivatives. The translation operator for δ is thus
and for δ′,
The difference between the two paths is then
It can be argued7 that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
where R is the Riemann curvature tensor and the sign depends on the sign convention of the author. Then the derivative of
f
(
S
)
{\displaystyle f({\boldsymbol {S}})}
with respect to
S
{\displaystyle {\boldsymbol {S}}}
(or at
S
{\displaystyle {\boldsymbol see page
) in the direction
T
{\displaystyle {\boldsymbol {T}}}
is the second order tensor defined as
Properties:
Let
F
(
S
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}
be a second order tensor valued function of the second order tensor
S
{\displaystyle {\boldsymbol {S}}}
.
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.