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Multivariable Calculus Gradient, Divergence, Curl cheat sheet - grade college

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Calculus Grade college

Multivariable Calculus Gradient, Divergence, Curl Cheat Sheet

A printable reference covering gradients, directional derivatives, divergence, curl, Laplacians, and vector identities for college students.

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This cheat sheet covers the three core differential operators of multivariable calculus: gradient, divergence, and curl. These tools describe how scalar fields and vector fields change in space, which is essential in physics, engineering, and advanced calculus. Students need a quick reference because the notation is compact, the meanings are geometric, and the formulas are easy to mix up. It is designed to help connect computation with interpretation.

Key Facts

  • For a scalar field f(x,y,z)f(x,y,z), the gradient is ∇f=⟨∂f∂x,∂f∂y,∂f∂z⟩\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle.
  • The directional derivative of ff in the unit direction u\mathbf{u} is Duf=∇f⋅uD_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}.
  • The gradient ∇f\nabla f points in the direction of greatest increase of ff, and the maximum rate of increase is ∥∇f∥\lVert \nabla f \rVert.
  • For a vector field F=⟨P,Q,R⟩\mathbf{F}=\langle P,Q,R\rangle, the divergence is ∇⋅F=∂P∂x+∂Q∂y+∂R∂z\nabla \cdot \mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}.
  • For a vector field F=⟨P,Q,R⟩\mathbf{F}=\langle P,Q,R\rangle, the curl is ∇×F=⟨∂R∂y−∂Q∂z,∂P∂z−∂R∂x,∂Q∂x−∂P∂y⟩\nabla \times \mathbf{F}=\left\langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right\rangle.
  • The Laplacian of a scalar field is ∇2f=∇⋅(∇f)=∂2f∂x2+∂2f∂y2+∂2f∂z2\nabla^2 f = \nabla \cdot (\nabla f)=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}.
  • If F=∇f\mathbf{F}=\nabla f, then ∇×F=0\nabla \times \mathbf{F}=\mathbf{0} whenever the needed second partial derivatives are continuous.
  • For sufficiently smooth vector fields, ∇⋅(∇×F)=0\nabla \cdot (\nabla \times \mathbf{F})=0.

Vocabulary

Scalar field
A scalar field assigns one number, such as temperature or height, to each point in space.
Vector field
A vector field assigns a vector, such as velocity or force, to each point in space.
Gradient
The gradient is the vector of first partial derivatives of a scalar field and points toward greatest increase.
Divergence
Divergence measures the net outward flow or source strength of a vector field at a point.
Curl
Curl measures the local rotation or circulation tendency of a vector field at a point.
Laplacian
The Laplacian is the divergence of the gradient and measures how a scalar field compares to nearby values.

Common Mistakes to Avoid

  • Using the gradient on a vector field is wrong because ∇f\nabla f is defined for a scalar field ff, while vector fields use operators such as ∇⋅F\nabla \cdot \mathbf{F} and ∇×F\nabla \times \mathbf{F}.
  • Forgetting that Duf=∇f⋅uD_{\mathbf{u}}f = \nabla f \cdot \mathbf{u} requires u\mathbf{u} to be a unit vector is wrong because a nonunit direction scales the derivative by its length.
  • Mixing up divergence and curl is wrong because ∇⋅F\nabla \cdot \mathbf{F} produces a scalar, while ∇×F\nabla \times \mathbf{F} produces a vector in three dimensions.
  • Reversing signs in the curl formula is wrong because the component order determines orientation, especially in ∂P∂z−∂R∂x\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x} and ∂Q∂x−∂P∂y\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}.
  • Assuming zero divergence means zero curl is wrong because divergence describes source behavior, while curl describes rotational behavior.

Practice Questions

  1. 1 Find ∇f\nabla f for f(x,y,z)=x2y+yz3f(x,y,z)=x^2y+yz^3.
  2. 2 Compute ∇⋅F\nabla \cdot \mathbf{F} for F(x,y,z)=⟨x2,xy,z3⟩\mathbf{F}(x,y,z)=\langle x^2,xy,z^3\rangle.
  3. 3 Compute ∇×F\nabla \times \mathbf{F} for F(x,y,z)=⟨yz,xz,xy⟩\mathbf{F}(x,y,z)=\langle yz,xz,xy\rangle.
  4. 4 Explain why a field with ∇⋅F=0\nabla \cdot \mathbf{F}=0 can still have nonzero curl, using the meanings of divergence and curl.