An insect is moving on the ellipse $2x^2+y^2=3$ on the $xy$-plane in the clockwise direction at a constant speed of 3 centimeter per second. The temperature function $T(x,y)$ (experienced by the insect) is given by $T(x,y)=3x^2−2yx$, where $T$ is measured in degree Celsius and $x$, $y$ are measured in centimeters. What is the rate of change of the temperature (in degree Celsius per second) when the insect is at the point $(1,1)$?
Hint: Let $f(x;y)=2x^2+y^2$. The gradient vector of $(1,1) = \langle4, 2 \rangle$ is normal to the ellipse $f(x,y)=3$ at the point $(1,1)$. Using this information, how can we easily find a vector which is tangential to the ellipse $f(x,y)=3$ and is pointing in the clockwise direction?
My attempt to this question is firstly, since the gradient vector is a normal vector to the ellipse, I can use the property of it being a normal vector, such that the tangential vector, represented by $t$, $n \cdot t = 0$. However, is it possible to just use any values such that $tx$ and $ty$ can fulfill this property? After which, how can I proceed from here!
Best Answer
Let $\vec{v}=\langle4,2\rangle$. Then a vector orthogonal to $\vec{v}$ can be found by interchanging the components and changing one of the signs, so we can take $\vec{t}=\langle2,-4\rangle$ to get a tangent vector corresponding to the clockwise direction; and normalizing this vector gives the unit tangent vector $\vec{u}=\langle\frac{1}{\sqrt{5}},-\frac{2}{\sqrt{5}}\rangle$.
Now find the directional derivative of T in the direction of $\vec{u}$ using
$\;\;\;D_{\vec{u}}T(1,1)=\vec{\nabla T}(1,1)\cdot\vec{u}$