Question

The temperature at a point $(x,y,z)$ in the ball $x^2+y^2+z^2\le 1$ is given by $T=y^2+xz$ Find the largest and smallest values which $T$ takes (a) on the circle $y=0,x^2+z^2=1,$ (b) on the surface $x^2+y^2+z^2=1,$ (c) in the whole ball.

Short answer

Largest value of T: 1, smallest value of T: -1.

Step-by-step solution

- Understanding the problem

The temperature function is given by $T=y^2+xz$. The goal is to find the largest and smallest values that this function can take:

- Analyzing part (a)

On the circle $y=0,x^2+z^2=1$:the temperature function simplifies to $T=xz$. Given $x^2+z^2=1$, we can use the parametrization $x=\cos (t),z=\sin (t)$. Then $T=\cos (t)\sin (t)=\frac{1}{2}\sin (2t)$. The extreme values of \sin(2t)\ range from -1 to 1, so \frac{1}{2}\sin(2t) \ ranges from -\frac{1}{2}\ to \frac{1}{2}\. Thus, for part (a), the largest value is \frac{1}{2}\, and the smallest value is -\frac{1}{2}\.

- Analyzing part (b)

On the surface $x^2+y^2+z^2=1$:We use Lagrange multipliers function $g(x,y,z)=x^2+y^2+z^2-1$ to find the extreme values. Setting \ \abla T = \lambda \abla g\, we get the system of equations:1. \ 2y = 2\lambda y \ 2. 2 = 2x\lambda \3. \ x = 2\lambda z\ 4. \ x^{2}+y^{2}+z^{2}=1\.For \ y eq 0, \ \lambda = 1\ so \ 2\lambda = 0, 2x \lambda = 2 x, x = 2 \lambda z}}}5.\ y=0 xz = \ +-1 Any Point x2+z2 outside\ thus inconsiderably disregard since on circle\

- Analyzing part (c)

In the whole ball $x^2+y^2+z^2\le \le 1$: Since we know that the temperature function $T=y^2+xz$ is continuous and we are dealing with a closed, bounded subset, the function will attain its absolute maximum and minimum values on this set. Bringing observations from parts (a) and (b) together, the absolute max value would be \frac*{1}{1}=1 and least value function = -1 on surface\.

Key concepts

Temperature Distribution

The temperature distribution in a given space tells us how temperature varies at different points. For the problem at hand, the temperature function is given as $T=y^2+xz$, where $x$, $y$, and $z$ are coordinates of a point within a ball defined by $x^2+y^2+z^2\le 1$.

This means that as you move from one point to another within this sphere, the temperature changes based on the values of $x$, $y$, and $z$. Understanding this distribution helps us identify where the hottest and coldest points are.
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For part (a), we look at the circle where the equation $y=0$ and $x^2+z^2=1$ hold true. Here, the temperature distribution simplifies to $T=xz$.

For part (b), we analyse the surface described by the equation $x^2+y^2+z^2=1$.

For part (c), we consider the entire ball defined by the inequality $x^2+y^2+z^2\le 1$.

Lagrange Multipliers

Lagrange multipliers is a strategy used to find the local maxima and minima of a function subject to equality constraints.

In part (b) of the problem, we use Lagrange multipliers to find the extreme values of the temperature function on the surface $x^2+y^2+z^2=1$.
The steps involved are:

• Define a constraint function, $g(x,y,z)=x^2+y^2+z^2-1$, which represents the surface.
• Find the gradients of the temperature function, $ablaT$, and the constraint function, $ablag$.
• Set up the system of equations by equating $ablaT=\lambda ablag$, where $\lambda$ is a scalar known as the Lagrange multiplier.
• Solve this system of equations along with the constraint $x^2+y^2+z^2=1$ to find the values of $x$, $y$, and $z$ that lead to the extreme temperature values.

Through these steps, we determine the largest and smallest values the temperature takes on the surface of the sphere.

Parametric Equations

Parametric equations represent a set of equations that express the coordinates of the points of a geometric object as functions of a variable, often called a parameter.

In part (a) of the problem, parametric equations simplify the temperature function on the circle $y=0$ and $x^2+z^2=1$.

By parameterizing, we let:

• $x=\cos (t)$
• $z=\sin (t)$

The temperature function becomes $T=\cos (t)\sin (t)=\frac{1}{2}\sin (2t)$.
This transformation allows us to easily find the range of the function and hence determine its extreme values.

Closed Subset Analysis

Closed subset analysis helps us identify that on a closed and bounded set, a continuous function will attain its maximum and minimum values.

In part (c) of the problem, we analyze the entire ball defined by $x^2+y^2+z^2\le 1$. The temperature function $T=y^2+xz$ is continuous, and the set defined by the ball is closed and bounded.

This ensures that the temperature function will attain an absolute maximum and minimum value within the ball. By analyzing the results from parts (a) and (b), we conclude that the extreme values within the ball are -1 and 1.
The closed subset analysis plays a critical role in determining that these extreme values are not just local, but absolute, within the defined region.