Question

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. $$w=f(x,y,z)=x{y}^2+x^{2}z+y{z}^2$$

Short answer

Question: Find the differential dw of the function $$w=f(x,y,z)=x{y}^2+x^{2}z+y{z}^2$$ in terms of the differentials of the independent variables x, y, and z. Answer: The differential dw is given by $$dw=(y^2+2xz)dx+(2xy+z^2)dy+(x^2+2yz)dz$$.

Step-by-step solution

Compute the partial derivatives.

First, we need to compute the partial derivatives of the function f(x, y, z) with respect to x, y, and z. These are as follows: Partial derivative with respect to x: $$\frac{\partial w}{\partial x}=\frac{\partial }{\partial x}(x{y}^2+x^{2}z+y{z}^2)=y^2+2xz$$ Partial derivative with respect to y: $$\frac{\partial w}{\partial y}=\frac{\partial }{\partial y}(x{y}^2+x^{2}z+y{z}^2)=2xy+z^2$$ Partial derivative with respect to z: $$\frac{\partial w}{\partial z}=\frac{\partial }{\partial z}(x{y}^2+x^{2}z+y{z}^2)=x^2+2yz$$

Compute the differential dw in terms of dx, dy, and dz.

Now, we use the partial derivatives to find the differential dw in terms of the differentials dx, dy, and dz. According to the multivariable chain rule, we have: $$dw=\frac{\partial w}{\partial x}dx+\frac{\partial w}{\partial y}dy+\frac{\partial w}{\partial z}dz$$ Substitute the partial derivatives that we have computed in step 1, and we get: $$dw=(y^2+2xz)dx+(2xy+z^2)dy+(x^2+2yz)dz$$ Therefore, the differential dw in terms of the differentials of the independent variables x, y, and z is: $$dw=(y^2+2xz)dx+(2xy+z^2)dy+(x^2+2yz)dz$$

Key concepts

Differential Calculus

Differential calculus is the branch of calculus that focuses on how functions change when their inputs change. It is all about understanding the concept of the derivative, which is a way to measure how fast something is changing at any given point. In our context, when dealing with multiple variables, differential calculus helps us explore how a function involving several variables changes when each of those variables is slightly varied. Understanding derivatives in differential calculus is essential to analyzing and predicting the changes in complex systems. It allows us to

• Determine rates of change
• Find slopes of tangent lines
• Analyze local extrema
• Model real-world phenomena

Differential calculus works hand in hand with integral calculus to form the fundamental backbone of mathematical analysis.

Partial Derivatives

When working with functions of more than one variable, we use partial derivatives. A partial derivative is a derivative where you hold some variables constant while differentiating with respect to another. For a function like $w=f(x,y,z)$, the partial derivative with respect to $x$ is found by treating $y$ and $z$ as constants.For example, the partial derivatives for our function given in the exercise are:

• With respect to $x$: $\frac{\partial w}{\partial x}=y^2+2xz$
• With respect to $y$: $\frac{\partial w}{\partial y}=2xy+z^2$
• With respect to $z$: $\frac{\partial w}{\partial z}=x^2+2yz$

The beauty of partial derivatives is that they allow us to understand how each variable influences the function on its own. This is crucial in fields such as physics and engineering, where multiple variables affect the behavior of systems.

Chain Rule

The chain rule is a fundamental concept in calculus that comes into play when dealing with composite functions. In multivariable calculus, a form of the chain rule helps compute the differential of a function in terms of its independent variables. The multivariable chain rule can be expressed as:$$dw=\frac{\partial w}{\partial x}dx+\frac{\partial w}{\partial y}dy+\frac{\partial w}{\partial z}dz$$This shows how the change in the function $w$ relates to small changes $dx$, $dy$, and $dz$ in variables $x$, $y$, and $z$. The partial derivatives serve as scaling factors that tell us how much $w$ changes with each variable's differential.The chain rule is especially useful in predicting the behavior of functions under simultaneous small changes and is foundational in optimization problems and sensitivity analysis.

Independent Variables

In calculus, independent variables are the inputs of the function that can vary freely. When we talk about finding the differential $dw$ of a function $w=f(x,y,z)$, $x$, $y$, and $z$ serve as independent variables.These variables are termed "independent" because they can change without being dependent on each other. This independence is crucial for partial derivatives: it allows us to assess the impact of each variable on the output separately.Understanding independent variables is key when:

• Modeling systems with multiple inputs
• Identifying the contribution of each variable
• Conducting experiments where variables can be controlled individually

By recognizing which variables are independent, we can accurately apply techniques like partial differentiation and leverage them to solve complex problems intuitively.