a. If the partial derivatives and are equal to , then is continuous at . | ||
b. The directional derivative, if it exists, is a finite real number. | ||
c. For all vectors and , with and , it holds that exists if, and only if, exists. | ||
d. If the restriction of to the -axis has a minimum at , then . |
a. Regardless of what is, the function is never a constant. | ||
b. If , then is constantly . | ||
c. The function is infinitely differentiable. | ||
d. If , then . |
a. The Jacobian of a function at a point can be a negative number. | ||
b. Any positive number can arise as the Jacobian at the origin of some function . | ||
c. If is invertible and differentiable at the origin, then the differential mapping at the origin is represented by a unique matrix. | ||
d. If is continuous and satisfies for all , then is a linear transformation. |
a. is differentiable at if, and only if, all partial derivatives of at exist. | ||
b. is differentiable at if, and only if, is continuous in a neighborhood of . | ||
c. is differentiable at if, and only if, is differentiable at . | ||
d. is differentiable at if, and only if, there exists a linear transformation such that is differentiable at . |
a. If is differentiable at , then . | ||
b. is always differentiable at any point. | ||
c. need not be continuous at the origin. | ||
d. If the homogeneity condition is satisfied, then must actually be an integer. |
a. . | ||
b. . | ||
c. . | ||
d. . |
a. , . | ||
b. , . | ||
c. , . | ||
d. , . |
a. If at every point the Jacobian of is non-zero, then the function is globally invertible. | ||
b. If at some point the Jacobian of is , then is not locally invertible at . | ||
c. If the Jacobian of is at infinitely many points, then can be locally invertible at most finitely many points. | ||
d. If the Jacobian of at every point is non-zero, then at every point the function is locally invertible. |
a. If each is meager, then the union is meager too. | ||
b. If for all the set is meager, then the union is meager too. | ||
c. If and is meager, then is meager too. | ||
d. If and is not meager, then is not meager either. |
a. If has a maximum at , then . | ||
b. If and , then can have a maximum at . | ||
c. If , , and , then does not have a global minimum or maximum. | ||
d. If does not have an extremum at and , then . |
a. If are independent and is continuous at , then any directional derivative can be determined from and . | ||
b. If is differentiable at , then any directional derivative can be determined from and . | ||
c. If is differentiable at , then any directional derivative can be determined from and if, and only if, the vectors are independent. | ||
d. Since there are infinitely many directions in which to compute directional derivatives, knowledge of any finite number of directional derivatives is never sufficient to determine all directional derivatives. |
a. If is locally invertible at every point in its domain, then the Jacobian is never . | ||
b. If is not locally invertible at some point in its domain, then the Jacobian at that point is . | ||
c. The Jacobian is either always positive or always negative. | ||
d. If the Jacobian of attains a positive value and a negative value, then it also attains the value . |
a. For every , the set in is not meager. | ||
b. The set is meager. | ||
c. If is meager, then the set obtained from by adding to it all accumulation points of is also meager. | ||
d. A closed set is never meager. |
a. compute partial derivatives of a function conditioned on excluding particular directions. | ||
b. find the largest value of where , conditioned on . | ||
c. find the inverse of a function when the Inverse Function Theorem applies but does not yield an unconditioned value. | ||
d. normalize the Jacobian, conditioned on the sign of the first partial derivatives. |
a. If is a linear transformation, then it is differentiable at all points. | ||
b. If is differentiable at a point, then its behavior in a small-enough neighborhood of that point is approximated by a linear transformation. | ||
c. If is differentiable and the Jacobian of is very large, then it is possible that the can be approximated at a point by two different linear transformations. | ||
d. The Jacobian of cannot ever reveal information about how the volume of sets changes when is applied. |
a. If is differentiable at the origin, then the restriction of to any straight line through the origin is continuous at the origin. | ||
b. If is differentiable at the origin, then the restriction of to any straight line through the origin is differentiable at the origin. | ||
c. If any restriction of to a line through the origin yields a continuous function, then it is still possible that is not differentiable at the origin. | ||
d. is differentiable at the origin if, and only if, the restriction of along any straight line through the origin yields a function which is differentiable at the origin. |
a. , provided that or is continuous at . | ||
b. if, and only if, and are continuous at . | ||
c. implies that is differentiable at . | ||
d. implies that is continuous at . |
a. is continuous at the origin. | ||
b. . | ||
c. is differentiable at the origin. | ||
d. has a minimum at the origin. |
a. If attains a local minimum at , then all directional derivatives of at are equal to . | ||
b. If attains a global minimum at , then, computed at , the equality must hold. | ||
c. If attains a local minimum at a point , subject to a constraint given by , then for all , it holds that computed at is . | ||
d. If attains a local minimum at a point , subject to a constraint given by a function , then for all that satisfy for all , it holds that computed at is . |
a. If are locally invertible at the origin, then the composite function is locally invertible at the origin. | ||
b. The theorem that a function is locally invertible at the origin, provided , is a special case of the Inverse Function Theorem. | ||
c. For every , there exists a function that is not locally invertible at any point of its domain. | ||
d. If and are such that is not locally invertible at the origin, then at least one of or cannot be locally invertible at the origin. |
a. If , then exists. | ||
b. If , then exists. | ||
c. If exists, then . | ||
d. If exists, then . |
a. The set of all rational numbers is a set of zero content. | ||
b. The set of all irrational numbers is a set of zero content. | ||
c. A countable union of sets of zero content is itself a set of zero content. | ||
d. Any finite set is a set of zero content. |
a. If is satisfied for infinitely many values , then . | ||
b. If is continuous on , then either or . | ||
c. If the set is of zero content, then . | ||
d. If is continuous on and satisfies for infinitely many points , then . |
a. . | ||
b. if , if , and otherwise . | ||
c. , when and , and in all other cases, . | ||
d. , when and , and in all other cases, . |
a. If is continuous in , then . | ||
b. If is Riemann integrable on , then for all , the function is Riemann integrable on . | ||
c. If for all , the function is Riemann integrable on , then is Riemann integrable on . | ||
d. If , then for all , it holds that , provided the integral exists. |
a. The integral does not exist. | ||
b. The integral exists and equals . | ||
c. The integral exists and equals . | ||
d. The integral exists, but its value can't be evaluated in closed form. |
a. The Jordan content of is never equal to the Jordan content of . | ||
b. If is invertible, then the Jordan content of is equal to the determinant of . | ||
c. If is invertible, then the Jordan content of is equal to the absolute value of the determinant of . | ||
d. The Jordan content of is equal to the absolute value of the determinant of . |
a. The union of finitely many Jordan measurable subsets is itself Jordan measurable. | ||
b. The intersection of a Jordan measurable subset and a set of zero content is itself a set of zero content. | ||
c. If and is not Jordan measurable, then is not Jordan measurable. | ||
d. If and is Jordan measurable, then is Jordan measurable. |
a. If is a linear transformation such that is itself a square of area . | ||
b. If for all . | ||
c. The stated equality never holds. | ||
d. The stated equality always holds. |
a. . | ||
b. . | ||
c. . | ||
d. . |
a. If holds for infinitely many points, then . | ||
b. If , then holds for all but finitely many points. | ||
c. If , then holds for infinitely many points. | ||
d. All of the answer options above hold true. |
a. If is not bounded, then is not Riemann integrable. | ||
b. If satisfies that for all , and is of zero content, then is Riemann integrable if, and only if, is Riemann integrable. | ||
c. If is continuous and positive at at least one point, then implies that attains negative values. | ||
d. All of the answer options above are correct. |
a. If is Jordan measurable, then so is . | ||
b. If is Jordan measurable, then so is the closure . | ||
c. If and are Jordan measurable, then so is . | ||
d. If and are Jordan measurable and , then is Jordan measurable. |
a. It transforms the region in the plane into an annulus in the plane, with the outer circle having radius . | ||
b. It transforms the region in the plane into an annulus in the plane, with the outer circle having radius . | ||
c. It transforms the region in the plane into an annulus in the plane, with the outer circle having radius . | ||
d. It transforms the region in the plane into an annulus in the plane, with the outer circle having radius . |
a. if either or is rational, and otherwise. | ||
b. , if both and , and otherwise. | ||
c. if , and if . | ||
d. All of the answer options above are Riemann integrable functions. |
a. If and , then . | ||
b. If , then . | ||
c. If , then the Jordan measure of and is the same. | ||
d. If and , then . |
a. , since . | ||
b. , since it is a union of sets of zero content (recall that is the -dimensional sphere). | ||
c. where , since is negligible. | ||
d. All of the answer options above describe sets of zero content. |
a. Both and . | ||
b. Either or . | ||
c. For all , either or . | ||
d. For all , both and . |
a. There exists an infinite subset such that is Riemann integrable on . | ||
b. If is integrable on a rectangle , then is bounded on . | ||
c. If is discontinuous, then there exists a rectangle such that is not Riemann integrable on . | ||
d. If is constantly , then is integrable on any rectangle. |
a. If is both continuous and integrable, then is a constant. | ||
b. is continuous if, and only if, is integrable. | ||
c. If is integrable, then is continuous. | ||
d. If is continuous, then is integrable. |
a. . | ||
b. . | ||
c. . | ||
d. . |
a. if, and only if, . | ||
b. If is odd, then . | ||
c. If is odd, then . | ||
d. if, and only if, both and are odd. |
a. . | ||
b. . | ||
c. . | ||
d. if, and only if, or . |
a. . | ||
b. . | ||
c. for all forms on . | ||
d. if, and only if, . |
a. If is exact, then is closed. | ||
b. If is closed, then is exact. | ||
c. If is closed and exact, then . | ||
d. The validity or invalidity of each answer option above depends on the values of and . |
a. . | ||
b. . | ||
c. . | ||
d. . |
a. . | ||
b. . | ||
c. . | ||
d. . |
a. . | ||
b. . | ||
c. If , then . | ||
d. If , then . |
a. If and are closed, then is exact. | ||
b. If and are exact, then is closed. | ||
c. If and are exact, then is exact. | ||
d. If and are closed, then is closed. |
a. . | ||
b. . | ||
c. . | ||
d. All of the answer options above lead to a closed form. |
a. | ||
b. | ||
c. | ||
d. |
a. If , then is exact. | ||
b. If and is closed, then is exact. | ||
c. If is closed and is closed, then . | ||
d. If is exact and is closed, then . |
a. . | ||
b. , provided is parametrized counterclockwise, and if is parametrized clockwise. | ||
c. . | ||
d. . |
a. If is the global angle form, then . | ||
b. If is the global angle form and is closed, then is an integral multiple of . | ||
c. If is closed and is an integer, then is the global angle form. | ||
d. All of the above answer options are correct. |
a. If is a -form and a degenerate -chain, then . | ||
b. The boundary of a degenerate -chain is a degenerate -chain. | ||
c. Every boundary is a cycle. | ||
d. All of the above statements are correct. |
a. Every cycle is a boundary. | ||
b. holds for at least one chain . | ||
c. A chain is a boundary if, and only if, it is a cycle. | ||
d. All of the statements above are incorrect. |
a. . | ||
b. . | ||
c. . | ||
d. None of the above equalities is correct. |
a. Stokes' Formula. | ||
b. Green's Formula. | ||
c. Gauss' Formula. | ||
d. The resulting formula has no particular name. |
a. Such a matrix represents a linear transformation whose values are independent of a particular choice of coordinates. Since volume depends on coordinates, the determinant must be . | ||
b. Such a matrix represents a linear transformation whose values are dependent on a particular choice of basis. Since volume depends on the basis, the determinant must be . | ||
c. The columns of the matrix, being dependent, determine a degenerate region in which thus has -dimensional volume equal to . | ||
d. All of the above statements are correct geometric explanations. |
a. The global angle form and the gradient. | ||
b. The global angle form and the Jacobian. | ||
c. The volume forms and the gradient. | ||
d. The volume forms and the Jacobian. |