What is the Jacobian matrix for the vector-valued function f([x; y]) = [$x + \sin (y)$; $y + \sin (x)$]?

Conditions

• The function is a vector-valued map from $R^2$ to $R^2$.
• The output components are $f_1(x,y) = x + \sin (y)$ and $f_2(x,y) = y + \sin (x)$.
• The matrix is evaluated at a specific point, such as (-2, 1), to describe local linear behavior.

Reasoning, step by step

1. Decompose the vector-valued function into two scalar component functions: $f_1(x, y) = x + \sin (y)$ and $f_2(x, y) = y + \sin (x)$.
2. Identify the input variables x and y, which will form the columns of the matrix.
3. Identify the output components $f_1$ and $f_2$, which will form the rows of the matrix.
4. Compute the partial derivative of $f_1$ with respect to x to fill the top-left entry.
5. Compute the partial derivative of $f_1$ with respect to y to fill the top-right entry.
6. Compute the partial derivative of $f_2$ with respect to x to fill the bottom-left entry.
7. Compute the partial derivative of $f_2$ with respect to y to fill the bottom-right entry.
8. Assemble these four partial derivatives into a 2x2 matrix.

Example

The video explicitly constructs the matrix by placing ∂$f_1$/∂x in the top-left, ∂$f_1$/∂y in the top-right, ∂$f_2$/∂x in the bottom-left, and ∂$f_2$/∂y in the bottom-right, labeling the result the 'Jacobian Matrix'.

Common misconceptions

• Confusing the order of rows and columns; rows correspond to outputs and columns to inputs.
• Thinking the Jacobian is a single number rather than a matrix.
• Assuming the Jacobian describes the global transformation rather than the local linear approximation near a specific point.

Watch the explanation

Explore next

• How do you evaluate the Jacobian matrix at the point (-2, 1)?
• What does the first column of the Jacobian matrix represent geometrically?
• Why is the Jacobian matrix called the local linear transformation?