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Gradient: Quiz


Question 1: If f and g are real-valued functions differentiable at a point aRn, then the ________ asserts that the product (fg)(x) = f(x)g(x) of the functions f and g is differentiable at a, and
Product ruleIntegralDifferential calculusDerivative

Question 2: This equation is equivalent to the first two terms in the multi-variable ________ expansion of f at x0.
Taylor's theoremTrigonometric functionsSeries (mathematics)Taylor series

Question 3: In ________ (Schey 1992, pp. 139–142):
Spherical coordinate systemCartesian coordinate systemCylindrical coordinate systemCoordinate system

Question 4: A (continuous) gradient field is always a conservative vector field: its ________ along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals).
DerivativeLine integralContinuous functionDifferential calculus

Question 5: Although the gradient is defined in term of coordinates, it is contravariant under the application of an ________ to the coordinates.
Orthogonal groupOrthogonal matrixGroup (mathematics)Matrix (mathematics)

Question 6: Let U be an ________ in Rn.
Topological spaceTopologyOpen setMetric space

Question 7: It follows that in this case the gradient of f is ________ to the level sets of f.
Vector spaceOrthogonalityEigenvalue, eigenvector and eigenspaceEuclidean vector

Question 8: For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20%, which is 40% times the ________ of 60°.
Generating trigonometric tablesTrigonometryPythagorean theoremTrigonometric functions

Question 9: A vector transforming in this way is known as a contravariant vector, and so the gradient is a special type of ________.
Metric tensorMultilinear algebraTensorExterior algebra

Question 10: where (Dg)T denotes the transpose ________.
Jacobian matrix and determinantDerivativeDifferential calculusMatrix calculus


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