Question

Find the path of a heat-seeking particle placed at the given point in space with a temperature field \(T(x, y, z)\). $$ T(x, y, z)=100-3 x-y-z^{2}, \quad(2,2,5) $$

Short answer

The path of the heat-seeking particle placed at (2,2,5) in the space with the temperature field \(T(x, y, z)\) is the gradient vector \(\nabla T|_{(2,2,5)} = -3\hat{i} - \hat{j} -10\hat{k}\)

Step-by-step solution

Understanding the Gradient Vector

The temperature gradient vector of the function \(T(x, y, z)\) is given by the operation \(\nabla T\), providing the direction of maximum temperature increase. To compute this, we need to find the partial derivatives of \(T\) with respect to \(x\), \(y\), and \(z\).

Computing Partial Derivatives

Compute the partial derivatives of the function \( T(x, y, z)\) with respect to \(x\), \(y\), and \(z\). Deriving, we get: \(\frac{\partial T}{\partial x} = -3\) \(\frac{\partial T}{\partial y} = -1\) \(\frac{\partial T}{\partial z} = -2z\) Hence, the gradient of \(T(x, y, z)\) is \(\nabla T = -3\hat{i} - \hat{j} -2z\hat{k}\)

Evaluate the Gradient at Given Point

The final task is to evaluate the gradient vector at the given point (2,2,5). Substituting \(x=2\), \(y=2\), \(z=5\) into \(\nabla T\), we get \(\nabla T|_{(2,2,5)} = -3\hat{i} - \hat{j} -2(5)\hat{k} = -3\hat{i} - \hat{j} -10\hat{k}\)

Conclusion

The path of a heat-seeking particle placed at (2,2,5) in the space with a temperature field \(T(x, y, z)\) would be the gradient vector \(\nabla T|_{(2,2,5)}\), which is \( -3\hat{i} - \hat{j} -10\hat{k}\). This indicates the direction of the maximum temperature increase at the point (2,2,5).\nSince the particle is heat-seeking, it will move in the path indicated by \(\nabla T\), seeking the zones of highest temperature.