← Back to 15. Multiple Integrals Calculus

📝 Transformations in the plane (14 MCQs)

📖 From Calculus • 15. Multiple Integrals Calculus • 14 questions available

←Previous Topic:Change of Variables in Multiple Integrals: Jacobians📝 Transformations in the planeJacobian determinant in two variables:Next Topic→

What is Transformations in the plane?

Definition:
A transformation $T(u,v) = (x,y)$ maps a region in the uv-plane to a region in the xy-plane. Common transformations include polar, linear, and nonlinear mappings.

Example:
The mapping $x=u+v, y=u-v$ transforms a square in uv-plane to a rotated square in xy-plane.

Reason:
Transformations simplify regions or integrands, making difficult integrals tractable by mapping them to simpler domains.

Easy

Medium

Hard

📝 All Transformations in the plane MCQs

Q1. A transformation is defined by $u=x+y$ and $v=x-y$ . Which expression correctly represents the absolute value of the Jacobian $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|$ ?

A.-1

B.-2

C.(1/2) ✅

D.-4

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Solving the transformation equations gives $x=(u+v)/2$ and $y=(u-v)/2$ . Therefore, the derivative matrix with respect to $u,v$ has determinant $-1/2$ , so its absolute value is $1/2$ . The reciprocal relationship between the two Jacobians is essential when changing variables in a double integral.

Q2. Why is a nonzero Jacobian determinant important when using a plane transformation locally?

A.It guarantees that every transformed region has equal area

B.It indicates that the transformation is locally invertible and does not collapse area into a lower-dimensional set ✅

C.It guarantees that the integrand becomes constant

D.It makes all transformed boundaries straight lines

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A nonzero Jacobian indicates that the transformation has locally independent coordinate directions. Consequently, nearby points are not collapsed into a curve or single point, and a local inverse can generally be formed. This property is what makes the change of variables meaningful for converting area elements and rewriting double integrals.

Q3. Suppose $u=x+y$ and $v=x-2y$ . A student claims that because both new variables contain $x$ , the transformation cannot be inverted. Which assessment is most accurate?

A.The student is correct because two equations cannot contain the same original variable

B.The student is correct because

x

must occur in only one transformed variable

C.The student is incorrect because the determinant of the transformation matrix is nonzero ✅

D.The student is incorrect only when

x=0

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The presence of the same original variable in both transformed coordinates does not prevent invertibility. The coefficient matrix is $\begin{pmatrix}1&1\\1&-2\end{pmatrix}$ , whose determinant is $-3$ , not zero. Hence the equations provide independent information and uniquely determine $x$ and $y$ from $u$ and $v$ .

Q4. A region is bounded by the lines $x+y=1$ , $x+y=4$ , $x-y=2$ , and $x-y=5$ . Which change of variables most directly converts this region into a rectangle?

u=x^2,\ v=y^2

u=x+y,\ v=x-y

✅

u=x/y,\ v=xy

u=x^2+y^2,\ v=x-y

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The boundary lines already have the forms $x+y=\text{constant}$ and $x-y=\text{constant}$ . Choosing $u=x+y$ and $v=x-y$ makes the four boundaries become $u=1$ , $u=4$ , $v=2$ , and $v=5$ . Thus the original slanted region maps directly to the rectangular region $1\leq u\leq4$ , $2\leq v\leq5$ .

Q5. A model uses $u=x+y$ and $v=xy$ to transform a region. At points where $x=y$ , the Jacobian of this transformation becomes zero. What is the best interpretation?

A.The transformation is guaranteed to preserve area there

B.The transformation becomes locally singular there, so the usual one-to-one change-of-variables argument may fail ✅

C.The transformed coordinates become identical everywhere

D.The original region must have zero area

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For $u=x+y$ and $v=xy$ , the Jacobian is $y-x$ , which vanishes when $x=y$ . A zero Jacobian means the coordinate directions become locally dependent, so the mapping can lose local invertibility. Therefore, applying the standard change-of-variables formula without addressing this singular set would be unjustified.

Q6. Consider $u=x^2-y^2$ and $v=2xy$ . A region in the first quadrant is transformed using these variables. Which feature should be checked before treating the transformed region as having a simple one-to-one correspondence with the original region?

A.Whether

u+v

is positive

B.Whether the Jacobian vanishes or the mapping identifies multiple points ✅

C.Whether

x+y

is constant

D.Whether the original axes are perpendicular

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The transformation resembles the real and imaginary parts of the square of a complex number, so multiple points can potentially map to related transformed coordinates. Its Jacobian is $4(x^2+y^2)$ , which is nonzero away from the origin, but global one-to-one behavior still depends on the chosen region. Thus both local nonsingularity and global uniqueness must be considered.

Q7. A rectangular region in the $uv$ -plane is mapped back by $x=(u+v)/2$ , $y=(u-v)/2$ . If the rectangle has area $12$ , what is the area of its image in the $xy$ -plane?

A.-3

B.-6 ✅

C.-12

D.-24

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The inverse transformation has absolute Jacobian $1/2$ . Therefore, each small area in the $uv$ -plane corresponds to half as much area in the $xy$ -plane. Assuming the transformation is one-to-one on the region, the image area is $12(1/2)=6$ . This illustrates why the Jacobian factor cannot be omitted.

Q8. A designer wants to transform the curved boundaries $xy=1$ , $xy=6$ , $x/y=2$ , and $x/y=5$ into constant-coordinate boundaries. Which variables are most appropriate?

u=x+y,\ v=x-y

u=x^2+y^2,\ v=x-y

u=xy,\ v=x/y

✅

u=x^2,\ v=y^2

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Each original boundary is already expressed through either the product $xy$ or the ratio $x/y$ . Setting $u=xy$ and $v=x/y$ turns the four boundaries into $u=1$ , $u=6$ , $v=2$ , and $v=5$ . This produces a rectangular description in the transformed coordinate system, provided the selected region avoids ambiguity.

Q9. A student transforms $u=x+y$ , $v=x-y$ , calculates $\frac{\partial(u,v)}{\partial(x,y)}=-2$ , and then multiplies the transformed integral by $-2$ . What is the student's main error?

A.The Jacobian should never be used in a double integral

B.The reciprocal Jacobian should be used, and area scaling requires its absolute value ✅

C.The Jacobian must always equal

1

D.The variables

u

and

v

must be squared before integration

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The calculated determinant $-2$ is the Jacobian from $x,y$ to $u,v$ , whereas the transformed area element requires $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|$ . The correct factor is therefore $1/|-2|=1/2$ . Using $-2$ both reverses orientation and applies the wrong magnitude of area scaling.

Q10. Another student argues that if the Jacobian is negative, the transformed integral must always have a negative value. Why is this reasoning incorrect?

A.A negative Jacobian only indicates orientation reversal; area scaling uses its absolute value ✅

B.A negative Jacobian means the original function is negative

C.The Jacobian affects only the limits, never the area element

D.Negative Jacobians are impossible for plane transformations

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The sign of a Jacobian determinant records orientation: a negative value means the mapping reverses local orientation. However, ordinary area is nonnegative, so the change-of-variables formula uses the absolute value of the relevant Jacobian. The sign of the determinant therefore does not by itself determine whether the integral is positive or negative.

Q11. A graph shows a family of parallel diagonal lines labeled by $u$ , and another family of parallel diagonal lines crossing them at right angles labeled by $v$ . A region is bounded by two lines from each family. Which conclusion is most reasonable?

A.The original region can be represented naturally by constant bounds on

u

and

v

✅

B.The transformation must preserve the original Cartesian coordinates

C.The Jacobian must equal zero because the boundaries are diagonal

D.The region cannot be described using a double integral

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: If the displayed boundary families correspond to $u=\text{constant}$ and $v=\text{constant}$ , then choosing those quantities as transformed coordinates converts the four-sided region into a coordinate rectangle. The visual structure indicates that the transformation is aligned with the geometry of the boundaries rather than the original $x$ - and $y$ -axes.

Q12. A computational model uses $u=x+y$ and $v=x-y$ to simplify a region, but the integrand contains $x^2+y^2$ . Which strategy is most efficient?

A.Keep

x^2+y^2

unchanged because transformed variables cannot simplify it

B.Solve for

x,y

in terms of

u,v

, rewrite the integrand, and include the Jacobian ✅

C.Change only the limits and leave the area element unchanged

D.Replace

x^2+y^2

u^2+v^2

without calculation

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A successful transformation requires changing more than just the limits. From $x=(u+v)/2$ and $y=(u-v)/2$ , one obtains $x^2+y^2=(u^2+v^2)/2$ , while the area factor is $1/2$ . Rewriting both the integrand and differential area consistently is necessary for an equivalent integral.

Q13. Suppose a transformation maps a small $xy$ -rectangle into a parallelogram whose area is approximately three times larger. What does this suggest about the local Jacobian magnitude at that point?

A.Approximately

1/3

for

\left|\frac{\partial(x,y)}{\partial(u,v)}\right|

B.Approximately

1

C.Approximately

3

for

\left|\frac{\partial(u,v)}{\partial(x,y)}\right|

only if orientation is preserved ✅

D.It must be zero

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A Jacobian determinant measures local area scaling. If the transformation from $x,y$ to $u,v$ expands area by a factor of approximately $3$ , then $\left|\frac{\partial(u,v)}{\partial(x,y)}\right|$ is approximately $3$ . The inverse Jacobian therefore has magnitude approximately $1/3$ . The distinction between forward and inverse transformations is crucial.

Q14. Two proposed transformations describe the same original region. Method A makes the boundaries rectangular but produces a complicated integrand. Method B produces a simple integrand but leaves curved, difficult bounds. Which choice is generally preferable for evaluating a double integral?

A.Always choose Method A because rectangular bounds are mandatory

B.Always choose Method B because simple integrands matter more

C.Compare the total simplification of both the integrand, Jacobian, and bounds before choosing ✅

D.Choose randomly because both methods must give different answers

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A good change of variables balances all parts of the transformed integral. A rectangular region is valuable, but a highly complicated integrand or Jacobian can eliminate that advantage. Conversely, simple algebra with difficult limits may also be inefficient. Comparing the transformed integrand, Jacobian, and bounds provides the soundest basis for selecting a method.

🔗 Related Topics (MCQs)

📝 Double Integrals in Calculus MCQs 📝 Volume under surface double integral MCQs 📝 Double integral definition Riemann sums MCQs 📝 How to evaluate double integrals MCQs 📝 Fubini's theorem for double integrals MCQs

←Previous Topic:Change of Variables in Multiple Integrals: Jacobians📝 Transformations in the planeJacobian determinant in two variables:Next Topic→