Let g
 be the function given by g(x,y)=(x^2+y^2,xy)
 for (x,y)∈R^2
. 
Let f
 be the function given by f(u,v)=(u+v,u^2,uv)
 for (u,v)∈R^2
. 
Consider the composite function f∘g
. 


find the Jacobian matrix Jf∘g(x,y)
. 

To find the Jacobian matrix \( J_{f \circ g}(x, y) \), we need to compute the Jacobian of the composite function \( f(g(x, y)) \), where the function \( g(x, y) = (x^2 + y^2, xy) \) and the function \( f(u, v) = (u + v, u^2, uv) \).

### Step 1: Jacobian of \( g(x, y) \)
We start by computing the Jacobian matrix of \( g(x, y) \). The Jacobian of \( g \) is the matrix of partial derivatives of \( g(x, y) \) with respect to \( x \) and \( y \):

\[
g(x, y) = \begin{pmatrix} x^2 + y^2 \\ xy \end{pmatrix}
\]

The Jacobian matrix \( J_g(x, y) \) is:

\[
J_g(x, y) = \begin{pmatrix}
\frac{\partial}{\partial x}(x^2 + y^2) & \frac{\partial}{\partial y}(x^2 + y^2) \\
\frac{\partial}{\partial x}(xy) & \frac{\partial}{\partial y}(xy)
\end{pmatrix}
= \begin{pmatrix}
2x & 2y \\
y & x
\end{pmatrix}
\]

### Step 2: Jacobian of \( f(u, v) \)
Next, we compute the Jacobian matrix of \( f(u, v) \). The function \( f(u, v) = (u + v, u^2, uv) \), so the Jacobian matrix \( J_f(u, v) \) is:

\[
J_f(u, v) = \begin{pmatrix}
\frac{\partial}{\partial u}(u + v) & \frac{\partial}{\partial v}(u + v) \\
\frac{\partial}{\partial u}(u^2) & \frac{\partial}{\partial v}(u^2) \\
\frac{\partial}{\partial u}(uv) & \frac{\partial}{\partial v}(uv)
\end{pmatrix}
= \begin{pmatrix}
1 & 1 \\
2u & 0 \\
v & u
\end{pmatrix}
\]

### Step 3: Chain Rule for the Jacobian of \( f \circ g \)
The Jacobian of the composite function \( f \circ g \) is given by the product of the Jacobians of \( f \) and \( g \):

\[
J_{f \circ g}(x, y) = J_f(g(x, y)) \cdot J_g(x, y)
\]

Substituting \( g(x, y) = (x^2 + y^2, xy) \) into the Jacobian matrix \( J_f(u, v) \), we get:

\[
J_f(g(x, y)) = J_f(x^2 + y^2, xy) = \begin{pmatrix}
1 & 1 \\
2(x^2 + y^2) & 0 \\
xy & x^2 + y^2
\end{pmatrix}
\]

### Step 4: Matrix Multiplication
Now, compute the matrix product:

\[
J_{f \circ g}(x, y) = \begin{pmatrix}
1 & 1 \\
2(x^2 + y^2) & 0 \\
xy & x^2 + y^2
\end{pmatrix}
\cdot
\begin{pmatrix}
2x & 2y \\
y & x
\end{pmatrix}
\]

Perform the multiplication:

\[
J_{f \circ g}(x, y) = \begin{pmatrix}
1(2x) + 1(y) & 1(2y) + 1(x) \\
2(x^2 + y^2)(2x) + 0(y) & 2(x^2 + y^2)(2y) + 0(x) \\
xy(2x) + (x^2 + y^2)(y) & xy(2y) + (x^2 + y^2)(x)
\end{pmatrix}
\]

Simplifying each term:

\[
J_{f \circ g}(x, y) = \begin{pmatrix}
2x + y & 2y + x \\
4x(x^2 + y^2) & 4y(x^2 + y^2) \\
2x^2y + y(x^2 + y^2) & 2xy^2 + x(x^2 + y^2)
\end{pmatrix}
\]

This is the Jacobian matrix \( J_{f \circ g}(x, y) \).

---

Would you like further details or clarifications on any part of the process?

Here are some related questions to deepen your understanding:

1. What is the significance of the Jacobian matrix in multivariable calculus?
2. How does the chain rule for Jacobians apply to general composite functions?
3. How would you find the determinant of the Jacobian matrix for a given function?
4. Can the Jacobian matrix tell you whether a function is invertible?
5. How do the partial derivatives in the Jacobian relate to the tangent plane at a point?

**Tip:** When multiplying matrices, always ensure the dimensions match up properly for matrix multiplication to be valid.

Let g be the function given by g(x,y) = (x^2 + y^2, xy) for (x, y) ∈ R^2. Let f be the function given by f(u,v) = (u + v, u^2, uv) for (u,v) ∈ R^2. Consider the composite function f ∘ g. Find the Jacobian matrix Jf ∘ g(x, y).

Discover how to compute the Jacobian matrix for the composite function f ∘ g, where g(x, y) = (x^2 + y^2, xy) and f(u, v) = (u + v, u^2, uv). This solution covers multivariable calculus concepts such as partial derivatives and matrix multiplication. Ideal for undergraduate calculus students.

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