Math Problem Statement
Consider the transformation given by: u=sin(x)+cos(y), v=cos(y)+sin(z), w=sin(z)+cos(x). Find the Jacobian.
Solution
To compute the Jacobian of the transformation , we start by defining the Jacobian matrix. The Jacobian matrix is a matrix of partial derivatives, given by:
\begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{bmatrix}.$$ ### Step 1: Compute Partial Derivatives #### For $$u = \sin(x) + \cos(y)$$: - $$\frac{\partial u}{\partial x} = \cos(x)$$ - $$\frac{\partial u}{\partial y} = -\sin(y)$$ - $$\frac{\partial u}{\partial z} = 0$$ #### For $$v = \cos(y) + \sin(z)$$: - $$\frac{\partial v}{\partial x} = 0$$ - $$\frac{\partial v}{\partial y} = -\sin(y)$$ - $$\frac{\partial v}{\partial z} = \cos(z)$$ #### For $$w = \sin(z) + \cos(x)$$: - $$\frac{\partial w}{\partial x} = -\sin(x)$$ - $$\frac{\partial w}{\partial y} = 0$$ - $$\frac{\partial w}{\partial z} = \cos(z)$$ ### Step 2: Form the Jacobian Matrix The Jacobian matrix becomes: $$J = \begin{bmatrix} \cos(x) & -\sin(y) & 0 \\ 0 & -\sin(y) & \cos(z) \\ -\sin(x) & 0 & \cos(z) \end{bmatrix}.$$ ### Step 3: Compute the Determinant of $$J$$ (Jacobian Determinant) The determinant of $$J$$ is calculated as follows: $$\text{det}(J) = \begin{vmatrix} \cos(x) & -\sin(y) & 0 \\ 0 & -\sin(y) & \cos(z) \\ -\sin(x) & 0 & \cos(z) \end{vmatrix}.$$ Using cofactor expansion along the first row: $$\text{det}(J) = \cos(x) \begin{vmatrix} -\sin(y) & \cos(z) \\ 0 & \cos(z) \end{vmatrix} - (-\sin(y)) \begin{vmatrix} 0 & \cos(z) \\ -\sin(x) & \cos(z) \end{vmatrix}.$$ 1. Compute the first minor: $$\begin{vmatrix} -\sin(y) & \cos(z) \\ 0 & \cos(z) \end{vmatrix} = (-\sin(y))(\cos(z)) - (0)(\cos(z)) = -\sin(y)\cos(z).$$ 2. Compute the second minor: $$\begin{vmatrix} 0 & \cos(z) \\ -\sin(x) & \cos(z) \end{vmatrix} = (0)(\cos(z)) - (-\sin(x))(\cos(z)) = \sin(x)\cos(z).$$ Substituting back: $$\text{det}(J) = \cos(x)(-\sin(y)\cos(z)) - (-\sin(y))(\sin(x)\cos(z)).$$ Simplify: $$\text{det}(J) = -\cos(x)\sin(y)\cos(z) + \sin(y)\sin(x)\cos(z).$$ Factorize: $$\text{det}(J) = \cos(z)(-\cos(x)\sin(y) + \sin(x)\sin(y)).$$ Simplify further: $$\text{det}(J) = \cos(z)\sin(y)(\sin(x) - \cos(x)).$$ ### Final Answer: $$\text{det}(J) = \cos(z)\sin(y)(\sin(x) - \cos(x)).$$ Would you like a detailed explanation of any specific step or related applications? --- ### Questions to Explore Further: 1. How does the Jacobian apply to coordinate transformations in physics? 2. Can the determinant of a Jacobian indicate if a transformation is invertible? 3. What happens if one of the variables is constant in this transformation? 4. How does the Jacobian simplify for simpler trigonometric transformations? 5. What is the significance of a zero Jacobian determinant in transformations? ### Tip: When computing Jacobians, ensure all partial derivatives are taken with respect to the correct variables to avoid sign errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Jacobian Matrix
Partial Derivatives
Determinants
Formulas
Jacobian matrix: J = [[∂u/∂x, ∂u/∂y, ∂u/∂z], [∂v/∂x, ∂v/∂y, ∂v/∂z], [∂w/∂x, ∂w/∂y, ∂w/∂z]]
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Jacobian Determinant Theorem
Suitable Grade Level
Undergraduate (Calculus III or Advanced Mathematics)
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