The temperature at a point on a metal plate is given by $T(x, y)=\frac{x}{x^{2}+y^{2}}$. WRITE VECTORS IN COMPONENT FORM TO 4 DECIMAL PLACES.
a) Find the direction of greatest increase of heat at the point $(4,3)$. $\vec{v}=$
b) Find the rate of maximum temperature increase. rate $=$
c) The direction of greatest decrease of at the point $(4,3)$ is
d) The rate of maximum decrease of temperature is
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Step 1 ：First, we need to find the partial derivatives of the function \(T(x, y)=\frac{x}{x^{2}+y^{2}}\) with respect to x and y. The partial derivative of T with respect to x is \(\frac{\partial T}{\partial x} = -2x^{2}/(x^{2} + y^{2})^{2} + 1/(x^{2} + y^{2})\) and the partial derivative of T with respect to y is \(\frac{\partial T}{\partial y} = -2xy/(x^{2} + y^{2})^{2}\).

Step 2 ：Next, we evaluate these partial derivatives at the point (4,3) to get the components of the gradient vector. The gradient of the function T at the point (4,3) is (-7/625, -24/625).

Step 3 ：The gradient vector points in the direction of the greatest rate of increase of the function. To find the direction of this vector, we normalize it by dividing each component by the magnitude of the vector. The magnitude of a vector (a, b) is given by \(\sqrt{a^{2} + b^{2}}\).

Step 4 ：Calculating the magnitude of the gradient vector, we get 1/25.

Step 5 ：Dividing each component of the gradient vector by the magnitude, we get the direction of the vector as (-0.2800, -0.9600).

Step 6 ：Thus, the direction of greatest increase of heat at the point (4,3) is \(\boxed{(-0.2800, -0.9600)}\).