Jacobian Calculator 3 Variable
Enter three functions of x, y, and z. Get symbolic partials, determinant value, and checks. Download results for lessons, transformations, and multivariable analysis today.
Enter three functions of x, y, and z. Get symbolic partials, determinant value, and checks. Download results for lessons, transformations, and multivariable analysis today.
Use x, y, and z as variables. Supported functions include sin, cos, tan, sqrt, log, exp, abs, pi, and e.
| Function Set | f(x,y,z) | g(x,y,z) | h(x,y,z) | Point |
|---|---|---|---|---|
| Trigonometric | x^2*y + sin(z) | y^2*z + cos(x) | z^2*x + exp(y) | (2, 1, 3) |
| Polynomial | x*y*z | x^2 + y^2 + z^2 | x + y - z | (1, 2, 3) |
| Transformation | r*sin(theta)*cos(phi) | r*sin(theta)*sin(phi) | r*cos(theta) | Use x=r, y=theta, z=phi |
For three functions f, g, and h with variables x, y, and z, the Jacobian matrix is:
J = ∂(f,g,h) / ∂(x,y,z)
J = [ [∂f/∂x, ∂f/∂y, ∂f/∂z], [∂g/∂x, ∂g/∂y, ∂g/∂z], [∂h/∂x, ∂h/∂y, ∂h/∂z] ]
The determinant is: det(J) = a(ei − fh) − b(di − fg) + c(dh − eg).
The absolute determinant gives the local volume scale factor.
A Jacobian matrix describes how a vector function changes. It is useful when several outputs depend on several inputs. This calculator works with three functions and three variables. It builds the full matrix of first partial derivatives. It also evaluates the matrix at a selected point.
The Jacobian is common in multivariable calculus. It appears in coordinate transformations, optimization, robotics, economics, physics, and engineering. A small change in x, y, or z may change every output. The matrix shows that relationship in a clear form.
Each row belongs to one output function. Each column belongs to one input variable. The first row shows the partial derivatives of f. The second row shows the partial derivatives of g. The third row shows the partial derivatives of h. This layout helps compare local sensitivity.
The determinant gives important geometric information. Its absolute value shows the local volume scaling factor. A positive value keeps orientation. A negative value reverses orientation. A value near zero suggests local compression. It may also show a singular transformation.
This tool shows symbolic derivatives first. Then it evaluates each derivative at the entered point. This makes the result easier to verify. It also gives a finite difference check. That check compares symbolic values with numerical estimates.
Use this calculator for homework, teaching, transformation checks, and applied modeling. It helps when manual differentiation becomes long. It is also helpful for checking determinant signs. Always review your entered expressions before using the result in final work.
A Jacobian matrix is a matrix of first partial derivatives. It shows how several output functions change with respect to several input variables.
Yes. This calculator is designed for three input variables. It uses x, y, and z and accepts three output functions.
The determinant shows local scaling. Its absolute value gives volume scale. Its sign shows whether orientation is preserved or reversed.
You can enter polynomial, trigonometric, logarithmic, exponential, and square root expressions. Use standard math syntax such as sin(x), log(x), and sqrt(x).
A zero determinant may mean the transformation is locally singular. It can also mean local volume collapses at the selected point.
Yes. It displays symbolic partial derivatives for every function and variable pair. You can also simplify the symbolic display.
The numeric check estimates derivatives using small changes. It helps compare symbolic derivatives with finite difference approximations.
Yes. You can download the current result as a CSV file or a PDF file using the export buttons.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.