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Vector Calculus cheat sheet - grade 12

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Calculus Grade 12

Vector Calculus Cheat Sheet

A printable reference covering gradients, divergence, curl, line integrals, surface integrals, and the major vector calculus theorems for grade 12.

TopicsCalculusVectorsDerivativesIntegralsStudy Guide
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Vector calculus studies functions and fields that vary across two-dimensional and three-dimensional space. This cheat sheet helps students connect vectors, derivatives, and integrals in a clear way. It is useful for understanding motion, fluid flow, electric fields, and other physical systems.

Grade 12 students need it to organize formulas that often look similar but mean different things.

The core ideas are the gradient, divergence, curl, line integrals, surface integrals, and flux. The gradient ∇f\nabla f points in the direction of fastest increase of a scalar field, while divergence ∇⋅F\nabla \cdot \mathbf{F} measures net outward flow. Curl ∇×F\nabla \times \mathbf{F} measures local rotation in a vector field.

The big theorems, including Green's Theorem, Stokes' Theorem, and the Divergence Theorem, connect integrals over regions to integrals over their boundaries.

Key Facts

  • The gradient of a scalar field f(x,y,z)f(x,y,z) 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 divergence of F=⟨P,Q,R⟩\mathbf{F}=\langle P,Q,R\rangle 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}.
  • The curl of F=⟨P,Q,R⟩\mathbf{F}=\langle P,Q,R\rangle 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.
  • A line integral of a vector field over a curve CC is ∫CF⋅dr=∫abF(r(t))⋅r′(t) dt\int_C \mathbf{F}\cdot d\mathbf{r}=\int_a^b \mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(t)\,dt.
  • Flux through an oriented surface SS is ∬SF⋅n dS\iint_S \mathbf{F}\cdot \mathbf{n}\,dS, where n\mathbf{n} is the chosen unit normal vector.
  • Green's Theorem states that ∮CP dx+Q dy=∬R(∂Q∂x−∂P∂y)dA\oint_C P\,dx+Q\,dy=\iint_R \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA for a positively oriented simple closed curve CC.
  • The Divergence Theorem states that ∬SF⋅n dS=∭E∇⋅F dV\iint_S \mathbf{F}\cdot \mathbf{n}\,dS=\iiint_E \nabla\cdot\mathbf{F}\,dV for a closed surface SS bounding a solid region EE.

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 ∇f\nabla f is a vector that points in the direction where a scalar field increases fastest.
Divergence
Divergence ∇⋅F\nabla \cdot \mathbf{F} measures how much a vector field spreads outward from a point.
Curl
Curl ∇×F\nabla \times \mathbf{F} measures the local rotation or swirling tendency of a vector field.
Flux
Flux measures how much of a vector field passes through a surface, usually written as ∬SF⋅n dS\iint_S \mathbf{F}\cdot\mathbf{n}\,dS.

Common Mistakes to Avoid

  • Using a non-unit direction vector in Duf=∇f⋅uD_{\mathbf{u}}f=\nabla f\cdot\mathbf{u} is wrong because the formula assumes u\mathbf{u} has length 11.
  • Confusing divergence and curl is wrong because ∇⋅F\nabla\cdot\mathbf{F} gives a scalar while ∇×F\nabla\times\mathbf{F} gives a vector in three dimensions.
  • Forgetting the orientation of a curve or surface is wrong because reversing orientation changes the sign of integrals like ∮CF⋅dr\oint_C \mathbf{F}\cdot d\mathbf{r} and ∬SF⋅n dS\iint_S \mathbf{F}\cdot\mathbf{n}\,dS.
  • Applying Green's Theorem to a curve that is not closed is wrong because ∮CP dx+Q dy=∬R(∂Q∂x−∂P∂y)dA\oint_C P\,dx+Q\,dy=\iint_R \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA requires a closed boundary.
  • Dropping the parameter derivative in a line integral is wrong because ∫CF⋅dr\int_C \mathbf{F}\cdot d\mathbf{r} must become ∫abF(r(t))⋅r′(t) dt\int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.

Practice Questions

  1. 1 Find ∇f\nabla f for f(x,y,z)=x2y+3yz2f(x,y,z)=x^2y+3yz^2 at the point (1,2,−1)(1,2,-1).
  2. 2 For F=⟨x2,yz,z3⟩\mathbf{F}=\langle x^2, yz, z^3\rangle, compute ∇⋅F\nabla\cdot\mathbf{F} at (2,1,−1)(2,1,-1).
  3. 3 Evaluate the line integral ∫CF⋅dr\int_C \mathbf{F}\cdot d\mathbf{r} for F=⟨y,x⟩\mathbf{F}=\langle y,x\rangle along r(t)=⟨t,t2⟩\mathbf{r}(t)=\langle t,t^2\rangle for 0≤t≤10\le t\le 1.
  4. 4 Explain why the Divergence Theorem applies only to closed surfaces and what physical meaning this has for net outward flow.