Math Problem Statement
Jacobians.pdf
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solve Q-5 on page 3
Solution
The question you referred to (Q-5 on page 3 of the document) is:
Q-5:
If and , find .
Solution:
We are asked to find the Jacobian of the functions and with respect to and , which is:
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}$$ Let's compute each of these partial derivatives step by step. #### Step 1: Compute $$\frac{\partial u}{\partial x}$$ and $$\frac{\partial u}{\partial y}$$ Given $$u = \frac{y}{x^2}$$, we compute the partial derivatives: - $$\frac{\partial u}{\partial x} = \frac{-2y}{x^3}$$ - $$\frac{\partial u}{\partial y} = \frac{1}{x^2}$$ #### Step 2: Compute $$\frac{\partial v}{\partial x}$$ and $$\frac{\partial v}{\partial y}$$ Given $$v = \frac{x^2 + y^2}{x^2} = 1 + \frac{y^2}{x^2}$$, we compute: - $$\frac{\partial v}{\partial x} = \frac{\partial}{\partial x} \left( 1 + \frac{y^2}{x^2} \right) = \frac{-2y^2}{x^3}$$ - $$\frac{\partial v}{\partial y} = \frac{\partial}{\partial y} \left( 1 + \frac{y^2}{x^2} \right) = \frac{2y}{x^2}$$ #### Step 3: Compute the Jacobian determinant The Jacobian matrix is: $$J(u, v) = \begin{vmatrix} \frac{-2y}{x^3} & \frac{1}{x^2} \\ \frac{-2y^2}{x^3} & \frac{2y}{x^2} \end{vmatrix}$$ The determinant is: $$J(u, v) = \left( \frac{-2y}{x^3} \times \frac{2y}{x^2} \right) - \left( \frac{1}{x^2} \times \frac{-2y^2}{x^3} \right)$$ Simplifying both terms: $$J(u, v) = \frac{-4y^2}{x^5} + \frac{2y^2}{x^5} = \frac{-2y^2}{x^5}$$ Thus, the Jacobian is: $$J(u, v) = \frac{-2y^2}{x^5}$$ ### Final Answer: $$\frac{\partial(u, v)}{\partial(x, y)} = \frac{-2y^2}{x^5}$$ Would you like more details on any step? Here are some related questions to explore: 1. How does the Jacobian relate to the concept of function independence? 2. What does it mean geometrically if the Jacobian is zero? 3. Can you explain the significance of a negative Jacobian determinant? 4. How would you solve this if the functions involved more complex terms like exponentials? 5. How can Jacobians be applied in real-world engineering problems? **Tip:** The Jacobian matrix plays a key role in transforming variables in multivariable calculus, especially in applications like changing coordinates in integrals.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Jacobian Determinant
Partial Derivatives
Multivariable Calculus
Formulas
Jacobian determinant: J(u, v) = det([[∂u/∂x, ∂u/∂y], [∂v/∂x, ∂v/∂y]])
Partial derivative: ∂f/∂x and ∂f/∂y
Theorems
Properties of Jacobians
Functional Independence and Dependence of Functions
Suitable Grade Level
Undergraduate