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Jacobian Matrix Calculator + Online Solver With Free Steps

 

A Jacobian Matrix Calculator is used to calculate the Jacobian matrix and other significant results from an input vector function.

The other resulting values from this calculator may include the Jacobian or also referred to as the Jacobian Determinant and the Jacobian Inverse.

The Jacobian and Jacobian Inverse both depend on the order of the Jacobian Matrix for their results and because of that, the order of the resulting matrix can change the results of this calculator by a lot.

This calculator can easily be used by entering the values in the input boxes.

What Is the Jacobian Matrix Calculator?

The Jacobian Matrix Calculator is a calculator which you can use online to solve for finding the Jacobian Matrix of your vector inputs. You can run this calculator easily in your browser and it can solve as many problems as you want.

A Jacobian Matrix tends to express the changes in the region around a function’s definition. This corresponds to a function’s transformation and its effects on its surroundings, and this has many applications in the field of engineering. 

Jacobian and its Matrix are both used for processes such as equilibrium predictions, map transformations, etc. A Jacobian Matrix Calculator helps in solving these quantities.

How To Use the Jacobian Matrix Calculator

The steps to use a Jacobian Matrix Calculator to the best of its abilities are as follows. You may want to start by setting up a problem for which you would like to calculate a Jacobian Matrix.

This calculator has two input boxes, one where you can enter your vector function in terms of $x$, $y$, etc., and the other where you enter your variables i.e., $x$, $y$, etc.

Now, follow the given steps to solve your Jacobian Matrix problem.

Step 1:

You will begin to enter the vector function with your concerned variables into the input box labeled “Jacobian Matrix of.”

Step 2:

You will follow that with the entry of the variables for your vector function in the input box labeled “concerning.”

Step 3:

Once you have entered both the input values, all that is left to do is to press the button labeled “Submit” and the calculator will solve the problem and show its results in a new window.

Step 4:

Finally, if you want to solve Jacobian Matrices for more problems you can simply enter your problem statements in this window and keep solving.

How Does the Jacobian Matrix Calculator Work?

The Jacobian Matrix Calculator works by performing first-order partial differentials on your given input problem. It also solves the determinant for this resulting matrix, which it can use to further find the inverse of the Jacobian Matrix.

Jacobian Matrix

A Jacobian Matrix is defined as the resulting matrix of the first-order partial derivative solution of a multivariable vector function. The significance of which lies in the study of differentials correlating with the transformation of coordinates.

To find a Jacobian Matrix, you first need a vector of functions of variables such as $x$, $y$ etc. The vector can be of the form $\begin{bmatrix} f_1(x, y, \ldots ) \\ f_2(x, y, \ldots) \\ \vdots   \end{bmatrix}$, where $ f_1(x, y, \ldots ) $, $ f_2(x, y, \ldots) $, and so on are both functions of $x$, $y$, and so on. Now, applying first-order partial differentials on this vector of functions can be expressed as:

\[\begin{bmatrix} \frac {\partial }{\partial x}f_1(x, y, \ldots) & \frac {\partial }{\partial y}f_1(x, y, \ldots) & \ldots \\ \frac {\partial }{\partial x}f_2(x, y, \ldots) & \frac {\partial }{\partial y}f_2(x, y, \ldots) & \ldots \\ \vdots & \vdots & \ddots  \end{bmatrix}\]

Jacobian

The Jacobian is another very important quantity associated with the vector of functions for a particular real-world problem. With its roots deep in the fields of Physics and Engineering, Jacobian is mathematically solved by finding the determinant of the Jacobian Matrix.

Thus, considering the generalized Jacobian Matrix we found above, we can calculate the Jacobian for it by using its determinant, where the determinant for a matrix of order $2 \times 2$ is given by:

\[ A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}\]

\[|A| =  \begin{vmatrix}a & b \\ c & d \end{vmatrix} = ad-bc\]

For order $3 \times 3$:

\[ A = \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\]

\[|A| =  \begin{vmatrix}a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a \cdot \begin{vmatrix}e & f \\ h & i\end{vmatrix} – b \cdot \begin{vmatrix}d & f \\ g & i\end{vmatrix} + c \cdot \begin{vmatrix}d & e \\ g & h\end{vmatrix}\]

\[|A| = a(ei – fh) – b(di – fg) + c(dh – eg)\]

Jacobian Inverse

The Jacobian Inverse is also exactly what it sounds like, which is the inverse of the Jacobian Matrix. The inverse of a matrix is calculated by finding the adjoint and the determinant of that matrix. The inverse of a matrix $A$ with the order $2 \times 2$ can be expressed as:

\[A^{-1} = \frac{Adj(A)}{|A|} = \frac{\begin{bmatrix}d & -b \\ -c & a \end{bmatrix}}{ad – bc}\]

Although the inverse of a $3 \times 3$ order matrix is more complicated as compared to the $2 \times 2$ order matrix, it can be calculated mathematically.

History of Jacobian Matrix

The concept of the Jacobian Matrix was introduced by the $19^{th}$ century mathematician and philosopher Carl Gustav Jacob Jacobi. This matrix is thus named after him as the Jacobian matrix.

The Jacobian Matrix was discovered as being the matrix resulting from taking first-order partial derivatives of the entries in a multivariable vector function. Ever since its introduction, it has been instrumental in the field of physics and mathematics where it is used for coordinate transformations.

Solved Examples

Here are some examples to look at.

Example 1

Consider the given vector $\begin{bmatrix}x+y^3 \\ x^3-y \end{bmatrix}$. Solve its Jacobian Matrix corresponding to $x$ and $y$.

We start by setting up the proper interpretation:

\[\begin{bmatrix}f_1 \\ f_2 \end{bmatrix} = \begin{bmatrix}x + y^3 \\ x^3 – y\end{bmatrix}\]

Now, solving for the Jacobian Matrix leads to:

\[\begin{bmatrix} \frac{\partial}{\partial x}f_1 & \frac{\partial}{\partial y}f_1\\ \frac{\partial}{\partial x}f_2 & \frac{\partial}{\partial y}f_2 \end{bmatrix} = \begin{bmatrix}\frac{\partial}{\partial x}(x + y^3) & \frac{\partial}{\partial y}(x + y^3)\\ \frac{\partial}{\partial x}(x^3 – y) & \frac{\partial}{\partial y}(x^3 – y) \end{bmatrix} = \begin{bmatrix}1 & 3y^2 \\ 3x^2 & -1\end{bmatrix}\]

The Jacobian determined is then expressed as:

\[\begin{vmatrix}1 & 3y^2 \\ 3x^2 & -1\end{vmatrix} = -9x^2y^2-1\]

Finally, the Jacobian Inverse is given as:

\[\begin{bmatrix}1 & 3y^2 \\ 3x^2 & -1\end{bmatrix}^{-1} = \begin{bmatrix} \frac{1}{9x^2y^2 + 1} & \frac{3y^2}{9x^2y^2 + 1} \\ \frac{3x^2}{9x^2y^2 + 1} & \frac{1}{-9x^2y^2 – 1}\end{bmatrix}\]

Example 2

Consider the given vector $\begin{bmatrix}x^3y^2-5x^2y^2 \\ y^6-3y^3 + 7 \end{bmatrix}$. Solve its Jacobian Matrix corresponding to $x$ and $y$.

We start by setting up the proper interpretation:

\[\begin{bmatrix}f_1 \\ f_2 \end{bmatrix} = \begin{bmatrix}x^3y^2-5x^2y^2 \\ y^6-3y^3 + 7\end{bmatrix}\]

Now, solving for the Jacobian Matrix leads to:

\[\begin{bmatrix} \frac{\partial}{\partial x}f_1 & \frac{\partial}{\partial y}f_1\\ \frac{\partial}{\partial x}f_2 & \frac{\partial}{\partial y}f_2 \end{bmatrix} = \begin{bmatrix}\frac{\partial}{\partial x}(x^3y^2-5x^2y^2) & \frac{\partial}{\partial y}(x^3y^2-5x^2y^2)\\ \frac{\partial}{\partial x}(y^6-3y^3 + 7) & \frac{\partial}{\partial y}(y^6-3y^3 + 7) \end{bmatrix} = \begin{bmatrix} 3x^2y^2-10xy^2 & 2x^3y-10x^2y \\ 0 & 6y^5-9y^2\end{bmatrix}\]

The Jacobian determined is then expressed as:

\[\begin{vmatrix}3x^2y^2-10xy^2 & 2x^3y-10x^2y \\ 0 & 6y^5-9y^2\end{vmatrix} = 3x(3x-10)y^4(2y^3-3)\]

Finally, the Jacobian Inverse is given as:

\[\begin{bmatrix}3x^2y^2-10xy^2 & 2x^3y-10x^2y \\ 0 & 6y^5-9y^2\end{bmatrix}^{-1} = \begin{bmatrix} \frac{1}{x(3x-10)y^2} & -\frac{2(x-5)x}{x(3x-10)y^3(2y^3-3)} \\ 0 & \frac{1}{6y^5-9y^2}\end{bmatrix}\]

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