Question

Suppose that \(u, v,\) and \(w\) are continuously differentiable functions of \((x, y, z)\) that satisfy the system $$ \begin{array}{l} e^{x} \cos y+e^{z} \cos u+e^{v} \cos w+x=3 \\ e^{x} \sin y+e^{z} \sin u+e^{v} \cos w \\ e^{x} \tan y+e^{z} \tan u+e^{v} \tan w+z=0 \end{array} $$ near \(\left(x_{0}, y_{0}, z_{0}\right)=(0,0,0),\) and \(u(0,0,0)=v(0,0,0)=w(0,0,0)=0 .\) Find \(u_{x}(0,0,0), v_{x}(0,0,0),\) and \(w_{x}(0,0,0)\)

Short answer

\(u(0,0,0)=v(0,0,0)=w(0,0,0)=0\) \(e^x\cos y + e^z\cos u + e^v\cos w + x = 3\) \(e^x\sin y + e^z\sin u + e^v\cos w = 0\) \(e^x\tan y + e^z\tan u + e^v\tan w + z = 0\) Answer: \(u_{x}(0,0,0) = 0, \quad v_{x}(0,0,0) = 0, \quad w_{x}(0,0,0) = 0\)

Step-by-step solution

Rewrite the system of equations at the given point

Since we are given that \(u(0,0,0)=v(0,0,0)=w(0,0,0)=0\) and we need to find the partial derivatives of u,v, and w at \((0,0,0)\). We can rewrite the system of equations at this point as: $$ \begin{array}{l} e^{0} \cos 0+e^{0} \cos 0+e^{0} \cos 0+0=3 \\\ e^{0} \sin 0+e^{0} \sin 0+e^{0} \cos 0 \\\ e^{0} \tan 0+e^{0} \tan 0+e^{0} \tan 0+0=0 \end{array} $$

Differentiate the equations with respect to x

Now, we will differentiate each equation with respect to x. For the first equation: $$ \frac{\partial}{\partial x}\left(e^x \cos y + e^z \cos u + e^v \cos w + x\right) = e^x\cos y + e^z(-\sin u)u_x + e^v(-\sin w)w_x + 1 $$ For the second equation: $$ \frac{\partial}{\partial x}\left(e^x \sin y + e^z \sin u + e^v \cos w\right) = e^x\sin y + e^z(\cos u)u_x + e^v(-\sin w)w_x $$ For the third equation: $$ \frac{\partial}{\partial x}\left(e^x \tan y + e^z \tan u + e^v \tan w + z\right) = e^x\sec^2 y + e^z\sec^2 u u_x + e^v\sec^2 w w_x $$

Evaluate the derivatives at the point (0,0,0)

Now that we have differentiated each equation with respect to x. We will evaluate the derivatives at the point (0,0,0): For the first equation: $$ 1 - u_x(0) - w_x(0) = 1 $$ For the second equation: $$ 0 + u_x(0) - w_x(0) = 0 $$ For the third equation: $$ 1 + u_x(0) + w_x(0) = 1 $$

Solve for the partial derivatives of u, v, and w

We can now solve this system of equations for the partial derivatives of u, v, and w at (0,0,0). Using the first and second equations, we get: $$ u_x(0) - w_x(0) = 0 $$ $$ u_x(0) = w_x(0) $$ Using the first and third equations we get: $$ 1 - u_x(0) - w_x(0) = 1 - 2u_x(0) $$ $$ u_x(0) = 0 $$ Since \(u_x(0) = w_x(0)\), then \(w_x(0) = 0\) as well. Finally, we get: $$ u_x(0,0,0) = 0, \quad v_x(0,0,0) = 0, \quad w_x(0,0,0) = 0 $$

Key concepts

Continuously Differentiable Functions

Continuously differentiable functions are an essential concept in calculus, particularly when dealing with multivariable functions. When we say a function is continuously differentiable, it means that not only does the function have derivatives, but these derivatives are also continuous.
This implies that the function, as well as its derivatives, do not exhibit any sudden jumps or breaks. This property is crucial for certain mathematical procedures, such as solving systems of equations or optimizing functions, to work as intended.
Continuously differentiable functions belong to the class of functions denoted as \(C^1\), indicating that first derivatives exist and are continuous. When analyzing problems involving several functions of multiple variables, like those in our original exercise, it's standard to assume the involved functions are continuously differentiable. This ensures that techniques such as partial differentiation can be applied smoothly without complications related to discontinuities.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In the context of this exercise, we are dealing with a system comprising functions \(u\), \(v\), and \(w\) which depend on variables \(x\), \(y\), and \(z\).
The goal often in such systems is to find values of these functions that satisfy all equations simultaneously.

• Each line in the system of equations represents a condition that needs to be met.
• By solving these, we can determine any dependencies or relationships between the functions.

In practical terms, systems of equations can model many real-world scenarios, such as points where curves in three-dimensional space intersect. In mathematical exercises like the one given, you typically solve for unknowns by substituting or manipulating equations to isolate variables. In this case, our solution involved finding the partial derivatives of \(u\), \(v\), and \(w\) at a specific point, which required understanding these dependencies.

Partial Derivatives

Partial derivatives are a fundamental concept when working with functions of multiple variables. They represent the rate of change of a function with respect to one of its variables, keeping the other variables constant.
In the exercise, we were concerned with finding partial derivatives, such as \(u_x\), \(v_x\), and \(w_x\), which denote how functions \(u\), \(v\), and \(w\) change with respect to \(x\) alone.

• This involves treating \(x\) as the only variable affecting the function, temporarily considering \(y\) and \(z\) as constants.
• Such derivatives provide insights into the function's behavior at a particular point.

In multivariable calculus, these are crucial because they extend the concept of a derivative to functions that depend on more than one variable. For practical purposes, these derivatives help determine how a system evolves, given small changes in one variable, while others remain fixed.In our specific solution, we differentiated the given system of equations with respect to one variable at a time (\(x\)), providing a clear method to solve for the desired values of \(u_x\), \(v_x\), and \(w_x\). Understanding how to calculate and interpret partial derivatives is vital for tackling problems like these.