Difficulty: Intermediate | Prerequisites: Partial derivatives, dot product, cross product, Sections 16.1 – 16.4
Vector calculus operators (gradient, curl, divergence) each act on a specific type of input and produce a specific type of output. Section 16.5 of Stewart's Multivariable Calculus is where you learn the rules for combining these operators and discover which combinations are mathematically valid. This matters because these identities underpin most of the major theorems coming in the rest of the course (Stokes' theorem, the divergence theorem) and because exams regularly test whether you can spot a meaningless expression. You should already be comfortable computing grad, curl, and div individually before tackling this material.
Gradient takes a scalar field and returns a vector field. Curl takes a vector field and returns a vector field. Divergence takes a vector field and returns a scalar field. When you chain them, you must check that the output of one is a valid input for the next, otherwise the expression is meaningless. Two key identities always hold (assuming sufficient smoothness): div(curl F) = 0 and curl(grad f) = 0.
Scalar field
A function f(x, y, z) that assigns a single real number to each point in space. Think of it as a temperature map: every location has one value.
Vector field
A function F(x, y, z) = ⟨P, Q, R⟩ that assigns a vector to each point in space. In simple terms, this is like a wind map where every location has a direction and magnitude.
Gradient (∇f or grad f)
An operator that takes a scalar field and returns a vector field: ∇f = ⟨∂f/∂x, ∂f/∂y, ∂f/∂z⟩. Think of it as pointing in the direction of steepest increase of f.
Curl (∇ × F or curl F)
An operator that takes a vector field and returns a vector field. It measures the local rotational tendency of the field. In simple terms, if you dropped a tiny paddle wheel into a fluid flow, curl tells you how fast and in which direction it would spin.
Divergence (∇ · F or div F)
An operator that takes a vector field and returns a scalar field: div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z. Think of it as measuring how much the field is "spreading out" or "compressing" at each point.
Laplacian (∇²f)
The divergence of the gradient: ∇²f = div(grad f) = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z². It is a scalar-to-scalar operator. In simple terms, it measures how the value at a point compares to the average of its neighbours.
The central idea is type-checking. Each operator requires a specific input type and produces a specific output type.
grad (∇): scalar field → vector field
curl (∇ ×): vector field → vector field
div (∇ ·): vector field → scalar field
When you chain operators, the output of the inner one must be a valid input for the outer one. If it is not, the expression is meaningless.
The homework walks through a dozen combinations. The pattern to recognise:
curl(grad f): grad f is a vector field, so curl can act on it. Meaningful, and always equals the zero vector (assuming continuous second partials).
grad(div F): div F is a scalar field, so grad can act on it. Meaningful, produces a vector field.
div(curl F): curl F is a vector field, so div can act on it. Meaningful, and always equals zero.
curl(scalar field): curl requires a vector field input. Meaningless.
grad(vector field): grad requires a scalar field input. Meaningless.
div(scalar field): div requires a vector field input. Meaningless.
div(div F): div F is a scalar field, and div needs a vector field. Meaningless.
cross product of scalar and vector: you cannot take the cross product of a scalar field and a vector field. Meaningless.
div(curl F) = 0
If F = ⟨P, Q, R⟩ has continuous second-order partial derivatives, then curl F is a vector whose divergence works out to zero by the equality of mixed partials. The homework demonstrates this by computing the divergence of a specific proposed curl and showing it equals −sin z, not zero, thereby proving no such F exists.
curl(grad f) = 0
Similarly, if f has continuous second partials, the curl of ∇f is the zero vector, again by equality of mixed partials.
These identities are converses of each other in a sense: a curl-free vector field (on a simply connected domain) is a gradient, and a divergence-free vector field (on all of R³) is a curl.
The homework proves these identities component-by-component. Let F = ⟨P, Q, R⟩ and G = ⟨L, M, N⟩, and let f be a scalar function.
25. div(F + G) = div F + div G: Divergence distributes over addition. Write out (P+L)_x + (Q+M)_y + (R+N)_z and separate.
26. curl(F + G) = curl F + curl G: Curl distributes over addition. Same component-by-component approach.
27. div(fF) = f div F + F · ∇f: The product rule for divergence. Expand ∂(fP)/∂x + ∂(fQ)/∂y + ∂(fR)/∂z using the ordinary product rule on each term, then regroup.
28. curl(fF) = f curl F + (∇f) × F: The product rule for curl. Expand each component of ∇ × (fF) using the product rule, then identify f curl F and (∇f) × F.
29. div(F × G) = G · curl F − F · curl G: Expand the cross product, take its divergence, then regroup the twelve terms.
30. div(∇f × ∇g) = 0: This follows from identity 29 applied to F = ∇f, G = ∇g, using the fact that curl(grad) = 0.
31. curl(curl F) = grad(div F) − ∇²F: The "BAC-CAB" identity for the del operator. The Laplacian ∇²F is applied component-wise. This is one of the most important identities in mathematical physics.
\nabla f = \left\langle \frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y},\; \frac{\partial f}{\partial z} \right\rangle\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\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\nabla \cdot (\nabla \times \mathbf{F}) = 0\nabla \times (\nabla f) = \mathbf{0}\text{div}(f\mathbf{F}) = f\,\text{div}\,\mathbf{F} + \mathbf{F} \cdot \nabla f\text{curl}(f\mathbf{F}) = f\,\text{curl}\,\mathbf{F} + (\nabla f) \times \mathbf{F}\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - \nabla^2 \mathbf{F}Students often think you can take the curl of any expression. You cannot: curl requires a vector field as input. Writing curl(f) where f is a scalar field is meaningless.
Students sometimes confuse div and grad. Gradient acts on scalars and returns a vector. Divergence acts on vectors and returns a scalar. They go in opposite directions.
A common error is assuming that div(curl F) equals something interesting. It is always zero (given smooth components). If you compute it and get a non-zero answer, you have made an algebra mistake.
Students frequently forget that curl(curl F) is not simply ∇²F. The correct identity is curl(curl F) = grad(div F) − ∇²F. The extra grad(div F) term matters.
⚠️ "Is this expression meaningful?" is a classic exam question. Know the input/output types cold.
⚠️ "Does there exist a vector field F such that curl F = [given expression]?" tests whether you remember to check div(curl F) = 0. If the divergence of the proposed curl is not zero, no such F exists.
⚠️ The product-rule identities (27, 28) are used in derivations throughout physics and engineering courses. Expect to see div(fF) or curl(fF) expansions on exams.
⚠️ The curl(curl F) = grad(div F) − ∇²F identity appears in electromagnetic theory and fluid dynamics. It frequently shows up on final exams.
True or false: div(grad f) is always zero. False – that is the Laplacian, ∇²f, which is generally non-zero. You are thinking of curl(grad f) = 0.
True or false: curl(curl F) = −∇²F. False – the correct identity is curl(curl F) = grad(div F) − ∇²F.
Fill in the blank: The divergence of the curl of any smooth vector field equals ___. Zero.
True or false: grad(div F) is meaningful. True – div F is a scalar, and grad acts on scalars.
True or false: div(div F) is meaningful. False – div F is a scalar, and you cannot take the divergence of a scalar.
Q: Determine whether the expression div(curl(grad f)) is meaningful. If so, what does it equal?
A: curl(grad f) = 0 (the zero vector field), and div(0) = 0. The expression is meaningful and equals zero.
Q: Suppose someone claims there exists a vector field F such that curl F = ⟨xeʸ, −eʸ + arcsin x / (1 + eᶻ), cos z + (1 + y²)^(1+x²)⟩. How would you test this claim?
A: Compute div(curl F). If F existed with continuous second partials, div(curl F) would have to equal zero. Computing the divergence of the given expression yields −sin z, which is not identically zero. Therefore no such F exists.
Q: Prove that div(fF) = f div F + F · ∇f.
A: Let F = ⟨P, Q, R⟩. Then fF = ⟨fP, fQ, fR⟩. By the product rule, ∂(fP)/∂x = f·∂P/∂x + P·∂f/∂x, and similarly for the y and z components. Collecting the f·(P_x + Q_y + R_z) terms gives f div F, and the remaining P·f_x + Q·f_y + R·f_z terms give F · ∇f.
Q: What is curl(curl F) in terms of other operators?
A: curl(curl F) = grad(div F) − ∇²F, where ∇²F applies the Laplacian to each component of F.
Q: Is the expression grad(div(curl F)) meaningful? What does it simplify to?
A: It is meaningful. div(curl F) = 0 (a scalar), and grad(0) = ⟨0, 0, 0⟩. The whole expression equals the zero vector.
The identity div(curl F) = 0 is the reason Stokes' theorem and the divergence theorem are consistent with each other. If a vector field G is the curl of something, then its flux through any closed surface is zero (by the divergence theorem, since div G = 0).
The identity curl(grad f) = 0 underpins conservative vector fields (Section 16.3): a gradient field is always irrotational, and on simply connected domains the converse holds too.
The curl(curl F) identity is the starting point for deriving the wave equation from Maxwell's equations in electromagnetism.
gradient, curl, divergence, del operator, nabla, vector calculus identities, div curl zero, curl grad zero, Laplacian, vector Laplacian, BAC-CAB identity, product rule for divergence, product rule for curl, meaningful vs meaningless expressions, Stewart 16.5, Calculus III, multivariable calculus, MATH 215, University of Michigan, Homework 9