∇Multivariable Calculus

Partial derivatives, gradient, Hessian, Laplacian, directional derivative, divergence, curl & tangent plane — with full step-by-step solutions.

f(x, y, z) =
f =
Variables:
∂/∂ Order:
Steps: Full working
Try these examples
∂f/∂y $x^2*y + sin(y)$ gradient $x^3 + y^3 - 3*x*y$ Hessian $x^2 + x*y + y^2$ Laplacian $sin(x*y)$ D_u f $x^2 + y^2$ tangent plane $exp(x+y)$
Steps Modes
None  Answer only.
Summary  Pills of all methods applied.
Detailed  Accordion with before→after, formula, and full explanation.
Operations
∂f/∂x
Partial Derivative
Rate of change w.r.t. one variable
∇f
Gradient
Vector of all partial derivatives
H(f)
Hessian
Matrix of 2nd partials; classifies critical points
∇²f
Laplacian
Sum of pure 2nd partials; curvature of field
D_û f
Directional
Rate of change in any direction û
∇·F
Divergence
Net outward flux; F solenoid if = 0
∇×F
Curl
Rotation of vector field; F conservative if = 0
z=z₀+…
Tangent Plane
Best linear approx to surface at (x₀,y₀)
Key Formulas
Chain Rule \(\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
Gradient \(\nabla f=(f_x, f_y, f_z)\)
Clairaut \(f_{xy}=f_{yx}\text{ (if continuous)}\)
2nd Deriv Test \(D=f_{xx}f_{yy}-f_{xy}^2\)
Laplacian \(\nabla^2f=f_{xx}+f_{yy}+f_{zz}\)
Directional \(D_\hat{u}f=\nabla f\cdot\hat{u}\)
Divergence \(\nabla\cdot F=P_x+Q_y+R_z\)
Curl i-comp \((\nabla\times F)_i=R_y-Q_z\)
Tangent Plane \(z=z_0+f_x(x-x_0)+f_y(y-y_0)\)
Hessian \(H_{ij}=\frac{\partial^2 f}{\partial x_i\partial x_j}\)
Input Tips
  • Scalar: x^2*y + sin(x*y)
  • Mixed partial: select order=2, choose mixed var
  • Directional: enter direction components (not unit — auto-normalized)
  • Divergence/Curl: enter P, Q, R in the component fields
  • Tangent plane: evaluation point is required
  • Gradient + eval point → numerical gradient vector
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