Problem 2 - Advanced Engineering Mathematics

Question

$$u=3 x^{2}-3 y^{2}+5 x, \quad v=6 x y+5 y-6 ; \quad \frac{\partial u}{\partial x}=6 x+5=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-6 y=-\frac{\partial v}{\partial x}$$

Step-by-Step Solution

Answer

The given conditions are satisfied: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).

Step 1: Understand the Exercise

The problem gives us two functions, \( u \) and \( v \), expressed in terms of variables \( x \) and \( y \). It also provides conditions given by partial derivatives: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). We need to verify whether these conditions hold true.

Step 2: Compute \( \frac{\partial u}{\partial x} \)

For \( u = 3x^2 - 3y^2 + 5x \), compute the partial derivative with respect to \( x \): \[ \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(3x^2) - \frac{\partial}{\partial x}(3y^2) + \frac{\partial}{\partial x}(5x) = 6x + 0 + 5 = 6x + 5. \]

Step 3: Compute \( \frac{\partial v}{\partial y} \)

For \( v = 6xy + 5y - 6 \), compute the partial derivative with respect to \( y \): \[ \frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(6xy) + \frac{\partial}{\partial y}(5y) - \frac{\partial}{\partial y}(6) = 6x + 5. \]

Step 4: Compare \( \frac{\partial u}{\partial x} \) and \( \frac{\partial v}{\partial y} \)

From the previous steps, we found \( \frac{\partial u}{\partial x} = 6x + 5 \) and \( \frac{\partial v}{\partial y} = 6x + 5 \). Thus, \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) holds true.

Step 5: Compute \( \frac{\partial u}{\partial y} \)

For \( u = 3x^2 - 3y^2 + 5x \), compute the partial derivative with respect to \( y \): \[ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(3x^2) - \frac{\partial}{\partial y}(3y^2) + \frac{\partial}{\partial y}(5x) = 0 - 6y + 0 = -6y. \]

Step 6: Compute \( \frac{\partial v}{\partial x} \)

For \( v = 6xy + 5y - 6 \), compute the partial derivative with respect to \( x \): \[ \frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(6xy) + \frac{\partial}{\partial x}(5y) - \frac{\partial}{\partial x}(6) = 6y + 0 = 6y. \]

Step 7: Compare \( \frac{\partial u}{\partial y} \) and \( -\frac{\partial v}{\partial x} \)

From the previous steps, we found \( \frac{\partial u}{\partial y} = -6y \) and \( \frac{\partial v}{\partial x} = 6y \). Therefore, \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \) also holds true.

Key Concepts

Function of Several Variables

When dealing with functions of several variables, such as in our exercise with the functions \( u(x, y) = 3x^2 - 3y^2 + 5x \) and \( v(x, y) = 6xy + 5y - 6 \), each variable contributes uniquely to the output of the function.
Such functions are foundational in multivariable calculus, where the variables \( x \) and \( y \) are considered independent, allowing us to explore varied rates of change.
Functions with several variables are depicted in multidimensional space. They are often essential in modeling physical systems where many inputs determine the behavior of an outcome.

• **Independent Variables**: Each independent variable influences the function's output and can change freely.
• **Partial Derivatives**: Key in exploring how changes in one variable impact the function, while other variables are held constant.
• **Applications**: Such functions frequently describe economic models, physical phenomena, or engineering problems.

Understanding how these variables interact is critical to predicting system behaviors in complex environments.

Conditions of Equality

In exercises involving partial derivatives, we often face conditions of equality, such as \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
Such conditions help establish relationships between the rates of change of different functions. In our scenario, these equations confirm that the partial derivatives meet specific criteria.
Understanding these conditions provides insight into the underlying symmetries or balance between the derivatives.

• **Verify Equality**: Compare the calculated derivatives to see if they satisfy the given equalities.
• **Symmetrical Relationships**: Often, these conditions point to symmetry or balance indicative of a system's broader behavior.
• **Problem Context**: In physics and engineering, these could relate to conservation laws or invariance under transformations.

Typically, satisfying conditions of equality confirms particular properties of functions, enhancing our models' accuracy and depth.

Verification of Derivatives

Verifying derivatives is a crucial step, ensuring that our calculations and logical interpretations are correct.
This practice involves rigorously checking that the computed derivatives meet the set conditions derived from the problem, as illustrated in our exercise.
Verification often involves comparing the results from each function's partial derivative with respect to different variables.

• **Calculate Accurately**: Carefully perform each derivation step to ensure accuracy.
• **Cross-Check Results**: Validate results by comparing them with provided conditions or relationships, ensuring they align.
• **Check Units and Context**: In some problems, especially physics-related, verifying ensures units or dimensions are consistent.

Consistently applying verification strengthens reliability and confidence in multivariable function analysis and can uncover deeper insights into a problem.