# Multivariable Calculus - Object movement and rate of change of its temperature problem

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) = <4, 2> 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?

Anyone please help solving this question! Even with the hint dropped by the lecturer, I still cannot get it!

You have the Gradient vector; this is the normal vector. To find the Tangent vector from the normal vector you can use the identity for the scalar product $nt = 0$ since normal vector $n$ is orthogonal to the Tangent vector $t$. This orthogonality also holds for non-normalized vectors. To find the components of the Tangent vector use the equation
$4t_x + 2t_y = 0$
Solve by one of the Parameters and you get the other Parameter to normalize the Tangent vector to $|t|=1$.
You finish your Task by computing the Gradient of the temperature $T(x,y)$ and by calculating the temperature Gradient projected to the normalized Tangent vector, i.e. $(gradT)t$.