Question

Using gradient rules Use the gradient rules of Exercise 85 to find the gradient of the following functions. $$f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}}$$

Short answer

Answer: The gradient vector of the given function is $$\nabla f = \left\langle \frac{-x}{\sqrt{25-x^2-y^2-z^2}}, \frac{-y}{\sqrt{25-x^2-y^2-z^2}}, \frac{-z}{\sqrt{25-x^2-y^2-z^2}}\right\rangle$$.

Step-by-step solution

Find the partial derivative with respect to x

First, we need to find the partial derivative of the function with respect to x. To do this, we differentiate the function with respect to x, treating y and z as constants: $$\frac{\partial f}{\partial x}=\frac{\partial}{\partial x} \sqrt{25-x^{2}-y^{2}-z^{2}}$$ Using the chain rule, we get: $$\frac{\partial f}{\partial x} = \frac{-x}{\sqrt{25-x^2-y^2-z^2}}$$

Find the partial derivative with respect to y

Similarly, we will find the partial derivative of the function with respect to y, treating x and z as constants: $$\frac{\partial f}{\partial y}=\frac{\partial}{\partial y} \sqrt{25-x^{2}-y^{2}-z^{2}}$$ Using the chain rule, we get: $$\frac{\partial f}{\partial y} = \frac{-y}{\sqrt{25-x^2-y^2-z^2}}$$

Find the partial derivative with respect to z

Finally, we will find the partial derivative of the function with respect to z, treating x and y as constants: $$\frac{\partial f}{\partial z}=\frac{\partial}{\partial z} \sqrt{25-x^{2}-y^{2}-z^{2}}$$ Using the chain rule, we get: $$\frac{\partial f}{\partial z} = \frac{-z}{\sqrt{25-x^2-y^2-z^2}}$$

Write the gradient vector

Now that we have found all the partial derivatives with respect to x, y, and z, we can write the gradient vector: $$\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle$$ Substitute the partial derivatives we found in the previous steps: $$\nabla f = \left\langle \frac{-x}{\sqrt{25-x^2-y^2-z^2}}, \frac{-y}{\sqrt{25-x^2-y^2-z^2}}, \frac{-z}{\sqrt{25-x^2-y^2-z^2}}\right\rangle$$ This is the gradient of the given function, $$f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}}$$.

Key concepts

Partial Derivatives

Understanding partial derivatives is a fundamental aspect of multivariable calculus, which involves functions with more than one variable. Imagine a function as a landscape over a two-dimensional grid. Just as a hiker might be interested in the steepness of the climb in a specific direction, a partial derivative represents how a function changes as one particular variable changes, while the other variables are held constant.

For the function f(x, y, z) = \( \sqrt{25-x^{2}-y^{2}-z^{2}} \), each partial derivative—\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, and \frac{\partial f}{\partial z}—gives us the rate of change in the directions of x, y, and z respectively. These derivatives are crucial for understanding the behavior of the function in different dimensions, each shedding light on how f varies with respect to one coordinate while freezing others.

Chain Rule

The chain rule is a powerful tool in calculus that allows us to differentiate composite functions—functions made up of other functions. It's like disassembling a machine to understand how its parts operate together. When we face a function like f(x, y, z) that involves a square root—a function within a function—the chain rule guides us through finding its derivative step by step.

In our exercise, the chain rule let us differentiate f with respect to x, y, and z by treating the square root as an outer function and the inner part 25 - x^2 - y^2 - z^2 as an inner function. By applying the chain rule systematically, we found that \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), and \( \frac{\partial f}{\partial z} \) are related to the derivatives of the inner function with regards to their respective variables.

Gradient Vector

The gradient vector is a central concept in understanding how a multivariable function increases most rapidly. If partial derivatives are individual hints to where a treasure might be, the gradient vector is the map pointing directly to the treasure—showing the path of steepest ascent. The gradient vector combines all the partial derivatives of a function into a single vector.

For the given function f(x, y, z), the gradient vector is denoted as \(abla f\) and is comprised of the partial derivatives we calculated: \( abla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle \) which tells us how the function f changes in a three-dimensional space. Each component of this vector indicates how fast the function is increasing in the direction of each corresponding axis.

Multivariable Calculus

In multivariable calculus, we go beyond the simple curves and surfaces of single-variable calculus to analyze surfaces, curves, and even higher-dimensional phenomena in multi-dimensional space. Studying functions of several variables, like \( f(x, y, z) \), requires different techniques than those of single-variable calculus because changes can happen in several directions at once.

In this field, concepts such as partial derivatives, the chain rule, and the gradient vector come together to give a richer understanding of functions that depend on more than one variable. Multivariable calculus allows us to evaluate rates of change, optimize functions over regions, and even model real-world phenomena in physics, engineering, economics, and beyond. Mastering this discipline helps us navigate the complexities of higher dimensions in both mathematics and applied sciences.