📝 Del operator ∇ in vector calculus (14 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 14 questions available

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What is Del operator ∇ in vector calculus?

Del operator $\nabla$ in vector calculus:
The del operator is a vector differential operator $\nabla = \langle \partial/\partial x, \partial/\partial y, \partial/\partial z \rangle$ , applied to scalar or vector fields.

Example:
Applied to scalar $f$ , $\nabla f = \langle f_x, f_y, f_z \rangle$ ; applied to vector $\mathbf{F}$ , $\nabla \cdot \mathbf{F}$ or $\nabla \times \mathbf{F}$ .

Reason:
Del unifies gradient, divergence, and curl, streamlining vector calculus notation and revealing underlying geometric meanings.

Easy

Medium

Hard

📝 All Del operator ∇ in vector calculus MCQs

Q1. For a scalar field $f(x,y,z)$ , suppose $f=x^2y+yz^2$ . Which interpretation of $?f$ is most appropriate when $?$ denotes the vector-calculus differential operator used to obtain the gradient?

A.A scalar measuring the net outward flow of

f

B.A vector containing the directional rates of change of

f

✅

C.A scalar measuring local rotation of

f

D.A vector tangent to every level surface of

f

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The gradient operator applied to a scalar field produces a vector containing the partial derivatives of the field with respect to the coordinate directions. It points toward the direction of greatest increase of $f$ . The other choices confuse gradient with divergence, curl, or tangent directions.

Q2. A student says that applying the same vector-calculus operator to a scalar field and to a vector field must always produce the same type of result. Which response best evaluates this claim?

A.It is correct because differential operators never change output type

B.It is correct only for constant fields

C.It is incorrect because the operator can produce different objects depending on whether it acts on a scalar or vector field ✅

D.It is incorrect only when the field contains three variables

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The operator commonly represented by $?$ is interpreted through its interaction with the object on which it acts. Applied to a scalar field, it produces a vector through the gradient. Applied through a dot product with a vector field, it produces a scalar divergence. Thus context matters.

Q3. Let $f(x,y)=x^2+4y^2$ . At the point $(1,-1)$ , a particle can move in any direction but wants $f$ to increase as rapidly as possible. Which vector should determine its instantaneous direction of motion?

(1,-4)

(2,-8)

✅

(2,8)

(-2,-8)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The gradient is $?f=(2x,8y)$ . At $(1,-1)$ , it becomes $(2,-8)$ , so the direction of greatest increase is represented by option B. The distractors arise from common mistakes such as changing the sign of the $y$ -component or incorrectly substituting the absolute value of the coordinate.

Q4. A vector field is given by $\mathbf F=(x^2,2xy,z^2)$ . If the operator $?$ is applied as a dot product with $\mathbf F$ , what quantity is obtained?

2x+2y+2z

2x+2x+2z

✅

x^2+2xy+z^2

2x+2x+2z^2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When $?$ acts as the divergence operator, it is evaluated as $? \cdot \mathbf F$ . Thus $? \cdot \mathbf F=\frac{\partial x^2}{\partial x}+\frac{\partial(2xy)}{\partial y}+\frac{\partial z^2}{\partial z}=2x+2x+2z$ . Careful differentiation with respect to the matching coordinate is required.

Q5. Consider a vector field $\mathbf F=(-y,x,0)$ . A student computes $? \times \mathbf F=0$ because the field has no $z$ -component. What is the fundamental error?

A.They ignored that derivatives are required when applying the operator ✅

B.They should first calculate the divergence

C.A zero

z

-component always makes curl zero

D.They should replace

x

and

y

by constants

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Curl depends on spatial derivatives of the vector components, not merely on whether one component is zero. For $\mathbf F=(-y,x,0)$ , the $z$ -component of the curl contains $\frac{\partial x}{\partial x}-\frac{\partial(-y)}{\partial y}=1-(-1)=2$ . Therefore the curl is not zero.

Q6. A velocity field is observed to have arrows that circulate counterclockwise around the origin, with stronger circulation closer to the center. Which use of the $?$ operator would best quantify the local rotational tendency?

A.Apply it as a gradient of a scalar speed

B.Apply it as a divergence

C.Apply it as a curl ✅

D.Apply it directly as a scalar multiplication

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A circulating vector-field pattern suggests local rotation. The curl operator is designed to measure this rotational tendency, with its direction indicating the local axis of rotation according to the right-hand rule. Divergence instead measures net local expansion or compression, while gradient applies naturally to scalar fields.

Q7. A temperature field is modeled by $T(x,y,z)=3x-2y+5z$ . An engineer wants to identify the direction in which temperature rises most rapidly from a given location. Which operation is most appropriate?

A.Compute

? \cdot \mathbf F

for the temperature field

B.Compute

?T

as a gradient ✅

C.Compute

?\times T

as a curl

D.Compute the divergence of

T

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Temperature is a scalar field, so the appropriate application of the differential operator is the gradient. Here $?T=(3,-2,5)$ , which is constant throughout space. This vector gives both the direction of maximum increase and the rate of maximum directional increase through its magnitude.

Q8. A flow field is modeled by $\mathbf F=(x,-y,2z)$ . At a point where $x=2$ , $y=1$ , and $z=-3$ , a student claims that the divergence must be negative because the third component of the field is negative. Which conclusion is correct?

A.The student is correct because

2z

is negative

B.The divergence depends on the field components themselves, not their derivatives

C.The divergence is positive because the coordinate derivatives sum to a positive value ✅

D.The divergence is zero because the field has both positive and negative components

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Divergence is determined by $\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$ , not by the signs of the components at one point. Here those derivatives are $1,-1,2$ , giving divergence $2$ . The negative value of $2z$ is irrelevant to that calculation.

Q9. A graph of a scalar field shows nested closed level curves centered at the origin, with values increasing toward the center. At a point on one of the curves, which direction should the vector produced by $?f$ point?

A.Along the level curve toward increasing arc length

B.Tangent to the level curve

C.Perpendicular to the level curve toward larger field values ✅

D.Perpendicular to the level curve toward smaller field values

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The gradient is perpendicular to level curves because moving tangentially along a level curve produces no first-order change in the scalar field. Since the graph indicates that values increase toward the center, the gradient at the selected point must point inward, perpendicular to the corresponding level curve.

Q10. A computational model gives a vector field $\mathbf F$ and reports both $? \cdot \mathbf F=0$ and $? \times \mathbf F=0$ throughout a simply connected region. Which interpretation is strongest?

A.The field must be zero everywhere

B.The field has neither local expansion nor local rotation in that region ✅

C.The field must have constant magnitude everywhere

D.The field must point radially outward everywhere

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A zero divergence indicates no local net expansion or compression, while a zero curl indicates no local rotational tendency. Having both zero does not imply that the vector field itself is zero or that its magnitude is constant. Additional boundary conditions would be needed to determine the field uniquely.

Q11. Suppose $\mathbf F=(yz,xz,xy)$ . A student evaluates $? \cdot \mathbf F$ as $yz+xz+xy$ . Why is this reasoning invalid?

A.The dot product is never allowed for vector fields

B.Divergence requires differentiating each component with respect to its corresponding coordinate ✅

C.The components must first be squared

D.The field must first be converted into a scalar

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Divergence is not the ordinary algebraic sum of vector components. The operator differentiates each component with respect to its corresponding coordinate: $? \cdot\mathbf F=\frac{\partial(yz)}{\partial x}+\frac{\partial(xz)}{\partial y}+\frac{\partial(xy)}{\partial z}$ . Each derivative is zero here, so the divergence is zero.

Q12. A surface is described by $f(x,y,z)=x^2+y^2+z^2$ . A particle constrained to move along a level surface of $f$ wants to avoid changing $f$ instantaneously. Which relationship should its velocity vector satisfy with $?f$ ?

A.It should be parallel to

?f

B.It should be perpendicular to

?f

✅

C.It should have the same magnitude as

?f

D.It should point opposite to

?f

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The gradient $?f$ is normal to the level surface. A velocity tangent to that surface therefore has zero dot product with the gradient, meaning $?f\cdot\mathbf v=0$ . This guarantees zero instantaneous directional change in $f$ , whereas a velocity parallel or antiparallel to the gradient changes the field most rapidly.

Q13. For $\mathbf F=(x^2y,xy^2,xyz)$ , an analyst wants to determine whether the field locally behaves like a source or sink at $(1,1,1)$ , and then determine whether it has local rotation there. Which sequence is mathematically appropriate?

A.Compute

?\times\mathbf F

first for source strength, then

?\cdot\mathbf F

for rotation

B.Compute

?\cdot\mathbf F

for source strength, then

?\times\mathbf F

for rotation ✅

C.Compute

?\mathbf F

once because both quantities are identical

D.Compute the gradient of each component and add the resulting vectors

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The divergence $?\cdot\mathbf F$ measures local source or sink behavior, while the curl $?\times\mathbf F$ measures local rotational tendency. For a vector field, these are different operations and generally produce different mathematical objects. Performing them in the stated sequence separates the two physical interpretations correctly.

Q14. Consider the scalar field $f(x,y,z)=x^2+y^2+z^2$ . At every nonzero point, the gradient points directly away from the origin. A student argues that this proves the gradient is tangent to every sphere centered at the origin. What is the best assessment?

A.Correct, because radial vectors lie on spherical surfaces

B.Correct only on the equator

C.Incorrect, because the gradient is normal to the sphere rather than tangent to it ✅

D.Incorrect, because the gradient has no geometric interpretation

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For $f=x^2+y^2+z^2$ , the gradient is $(2x,2y,2z)$ , which is radial. Spheres centered at the origin are level surfaces of $f$ , so their normals are radial. Therefore the gradient is perpendicular to each sphere, not tangent to it. This distinction is central to interpreting the operator geometrically.

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