Jacobian Matrix Calculator

How to Use the Jacobian Matrix Calculator

Step 1: Input the Number of Variables

Start by specifying the number of variables for your functions. This value, denoted as n, should be entered in the Number of Variables (n) field. This calculator accepts between 1 and 3 variables, though it is primarily set up for two-variable functions. Enter your chosen number of variables, making sure it is a whole number between 1 and 3.

Step 2: Select Functions

Next, choose the functions for which you want to compute the Jacobian matrix. For Function 1 (f₁), select one option from the dropdown list:

• x² + y
• sin(x) + cos(y)
• x·y
• eˣ + y

Similarly, select an option for Function 2 (f₂) from the following:

• x·y
• x² – y²
• eˣʸ
• ln(x+y)

Ensure that each function is correctly corresponding to the variables involved (usually x and y).

Step 3: Enter Variable Values

Provide the actual numerical values at which the Jacobian matrix should be evaluated. Input this data in the fields labeled x value and y value. These values denote the specific point (x, y) for which the partial derivatives will be calculated. The values can be decimals and should be input with a step size of 0.1.

Step 4: Calculate the Jacobian Matrix

With your functions and variable values set, the calculator automatically computes the partial derivatives:

• ∂f₁/∂x: The partial derivative of Function 1 with respect to x.
• ∂f₁/∂y: The partial derivative of Function 1 with respect to y.
• ∂f₂/∂x: The partial derivative of Function 2 with respect to x.
• ∂f₂/∂y: The partial derivative of Function 2 with respect to y.

These derivatives are displayed with a precision of four decimal places.

Step 5: Obtain the Determinant

Finally, the calculator provides the determinant of the Jacobian matrix. This value is computed using the formula:

Determinant = (∂f₁/∂x × ∂f₂/∂y) – (∂f₁/∂y × ∂f₂/∂x)

The determinant is displayed with four decimal places of accuracy, offering valuable insights into the nature of the functions’ interactions at the specified point.