Question

Calculate all first and second partial derivatives at \(\left(x_{0}, y_{0}\right)=(1,1)\) of the functions \(u\) and \(v\) that satisfy $$ \begin{array}{l} x^{2}+y^{2}+u^{2}+v^{2}=3 \\ x+y+u+v=3 \end{array} \quad u(1,1)=0, \quad v(1,1)=1 $$

Short answer

Based on the given conditions and the point (1,1), we have the following equations relating the partial derivatives of \(u\) and \(v\): 1. \(u_x + v_x = -1\) 2. \(u_y + v_y = -1\) 3. \(1+(u_x)^2+(v_x)^2+v_{xx}=0\) 4. \(1+(u_y)^2+(v_y)^2+v_{yy}=0\) 5. \(u_x+v_{xy}=0\) 6. \(u_y+v_{xy}=0\) These equations can't uniquely determine the exact values of the first and second derivatives of \(u\) and \(v\) at (1,1) with the given information in the problem. As a result, we are unable to find explicit values for each derivative at this point.

Step-by-step solution

Differentiate the conditions with respect to \(x\)

We will differentiate the given conditions with respect to \(x\). Condition 1: \(x^2 + y^2 + u^2 + v^2 = 3\) Differentiating with respect to \(x\), we get: \(2x + 2uu_x + 2vv_x= 0\) Condition 2: \(x + y + u + v = 3\) Differentiating with respect to \(x\), we get: \(1 + u_x + v_x = 0\)

Differentiate the conditions with respect to \(y\)

We will differentiate the given conditions with respect to \(y\). Condition 1: \(x^2+y^2+u^2+v^2=3\) Differentiating with respect to \(y\), we get: \(2y+2uu_y+2vv_y=0\) Condition 2: \(x+y+u+v=3\) Differentiating with respect to \(y\), we get: \(1+u_y+v_y=0\)

Differentiate with respect to \(x\) again for the second derivatives

Now we will differentiate the equations obtained in Steps 1 and 2 with respect to \(x\) again to find second derivatives. Dividing the first equation by 2 from step 1 (to simplify calculations), we get: \(x+u u_x+v v_x = 0\) Differentiating with respect to \(x\), we get: \(1+(u_x)^2+u u_{xx}+(v_x)^2+v v_{xx}=0\) Dividing the second equation by 2 from step 2 (to simplify calculations), we get: \(y+u u_y+v v_y=0\) Differentiating with respect to \(x\), we get: \(u_x+u u_{xy}+v v_{xy}=0\)

Differentiate with respect to \(y\) again for the second derivatives

Now we will differentiate the equations obtained in Steps 1 and 2 with respect to \(y\) again to find second derivatives. Differentiating the equation obtained in step 1 with respect to \(y\), we get: \(u_y+u u_{xy}+v v_{xy}=0\) Differentiating the equation obtained in step 2 with respect to \(y\), we get: \(1+(u_y)^2+u u_{yy}+(v_y)^2+v v_{yy}=0\)

Evaluate first and second partial derivatives at \((x_0, y_0) = (1,1)\)

Now we will evaluate the partial derivatives at the given point \((x_0, y_0) = (1, 1)\). Remember that we also have \(u(1,1)=0\), \(v(1,1)=1\). We will plug in these values into the equations obtained in Steps 1-4: 1. Step 1 - First derivatives with respect to \(x\) \(1 + u_x + v_x = 0\) Plugging in \(v(1,1) = 1\), we get: \(u_x + v_x = -1\) 2. Step 2 - First derivatives with respect to \(y\) \(1 + u_y + v_y = 0\) Plugging in \(v(1,1) = 1\), we get: \(u_y + v_y = -1\) 3. Step 3 - Second derivatives with respect to \(x\) \(1+(u_x)^2+u u_{xx}+(v_x)^2+v v_{xx}=0\) Plugging in \(u(1,1)=0\), \(v(1,1)=1\), we get: \(1+(u_x)^2+(v_x)^2+v_{xx}=0\) 4. Step 4 - Second derivatives with respect to \(y\) \(1+(u_y)^2+u u_{yy}+(v_y)^2+v v_{yy}=0\) Plugging in \(u(1,1)=0\), \(v(1,1)=1\), we get: \(1+(u_y)^2+(v_y)^2+v_{yy}=0\) 5. Equations from both Steps 3 and 4 \(u_x+u u_{xy}+v v_{xy}=0\) and \(u_y+u u_{xy}+v v_{xy}=0\) Plugging in \(u(1,1)=0\), \(v(1,1)=1\), we get: \(u_x+v_{xy}=0\) and \(u_y+v_{xy}=0\) From these equations, we can't find exact values for the first and second derivatives of \(u\) and \(v\) at \((1,1)\), but we have a system of equations that relate their values. The given information is not enough to uniquely determine the derivatives, so we can't find explicit values for each derivative in this case.

Key concepts

First Partial Derivative

Understanding the concept of the first partial derivative is vital in multivariable calculus and real analysis. These derivatives show how a function changes as each variable changes, while the other variables remain constant.

For a function of two variables, say, f(x, y), the first partial derivative with respect to x is denoted as f_x or ∂f/∂x. It is the derivative of the function with respect to x with y held constant. Similarly, f_y or ∂f/∂y is the derivative with respect to y keeping x fixed.

Returning to the provided exercise, the differentiating condition 2 with respect to x yields 1 + u_x + v_x = 0, which indicates the rate of change of u and v with respect to x. When evaluated at the point (1,1), it tells us how the functions u and v are inclined to change as x changes near this point.

Second Partial Derivative

The concept of a second partial derivative extends our understanding of curvature and concavity for functions with multiple variables. For the function f(x, y), the second partial derivatives are represented as f_xx, f_yy, and the mixed derivatives f_xy and f_yx.

These derivatives can be interpreted as the rate of change of the rate of change, or the 'acceleration' of the function with respect to one variable, or across the interaction of two variables. For example, f_xx is the partial derivative of f_x with respect to x, providing insight into the concavity in the x-directed slice of the function.

In the given exercise, differentiation of the simplified equation 1+(u_x)²+ u u_xx+(v_x)² + v v_xx=0 at the point (1,1) could shed light on the curvature of u and v if the derivatives were calculable at that point.

Multivariable Calculus

Multivariable calculus is the branch of mathematics dealing with functions of more than one variable. In this advanced field, we study topics like partial derivatives, multiple integrals, vector calculus, and their applications.

• Partial derivatives,
• Second partial derivatives, and
• Gradient vectors

are just a few of the key concepts dealt with in multivariable calculus.

It allows us to analyze the behavior of surfaces and spatial curves, optimize functions of several variables, and much more. In our textbook example, multivariable calculus is used to understand how two functions, u and v, interact and depend on two variables, x and y.

Real Analysis

The field of real analysis is a profound study of real numbers and real-valued functions. It constitutes the theoretical underpinning of calculus and deals with the properties of continuous, differentiable, and integrable functions.

Real analysis provides the tools for more rigorously understanding the concepts of limits, continuity, and convergence. Within the context of functions like those described in our exercise, real analysis would help us ensure that our functions are well-behaved and that our derivatives don't lead to contradictions or undefined behavior.

More advanced topics in real analysis, such as metric spaces, series convergence, and measure theory, prepare students for revealing and managing subtleties not immediately apparent in elementary calculus.