Vector Fields, Line & Surface Integrals – Core Vocabulary

30 vocabulary flashcards summarizing essential terms and facts about vector fields, line and surface integrals, divergence, curl, conservative fields, and related theorems discussed in the lecture notes.

True or False: A vector field is a function that assigns a vector to every point in a specific region of space.

True, a vector field is indeed a function that assigns a vector to every point in a specific region of space, often represented as F(x,y,z) = P i + Q j + R k in three dimensions.

True or False: A gradient vector field is always conservative and can be derived from a scalar potential function f(x,y,z).

True, a gradient vector field, by definition, is derived from a scalar potential function f(x,y,z) and is always conservative.

True or False: A vector field F is conservative if there exists a scalar function f such that F = ∇f, implying its line integrals are path-independent and circulation around any closed curve is zero.

True, a vector field F is defined as conservative if there exists a scalar function f such that F = ∇f. This key property results in path-independent line integrals and zero circulation around any closed curve.

True or False: A potential function (or scalar potential) is the scalar function f whose gradient, ∇f, produces a conservative vector field F.

True, the potential function (or scalar potential) is precisely that scalar function f whose gradient, ∇f, produces the given conservative vector field F.

True or False: The line integral of a vector field, represented as \intC F \cdot T ds or \intC F \cdot dr, measures the work done by the vector field F along a given curve C.

True, the line integral of a vector field, symbolized as \intC F \cdot T ds or \intC F \cdot dr, fundamentally quantifies the work done by the vector field F along a specified curve C.

True or False: Path independence is a property specific to conservative vector fields, meaning the line integral \int_C F \cdot dr depends solely on the starting and ending points, and not the specific path taken.

True, path independence is a defining characteristic of conservative fields; it means that the value of the line integral \int_C F \cdot dr is determined solely by the starting and ending points of the path, not the specific curve taken between them.

True or False: The closed-curve integral, also known as circulation, is the line integral of a vector field around a closed path, and for a conservative field, this integral always equals zero.

True, the closed-curve integral, often called circulation, is the line integral of a vector field around a closed path. For conservative fields, this integral is indeed always zero due to path independence.

True or False: Reversing the orientation of a curve, by traversing it in the opposite direction, results in multiplying its line integral by -1.

True, reversing the orientation of a curve, meaning traversing it in the opposite direction, always results in the line integral's value being multiplied by -1.

True or False: A flow line (or field line) is a curve where the tangent vector at every point aligns with the direction of the vector field at that point, and multiple distinct flow lines can exist within the same vector field.

True, a flow line (or field line) is precisely a curve where the tangent vector at every point aligns with the direction of the vector field at that point, and it's common for multiple distinct flow lines to belong to the same vector field.

True or False: The surface integral of flux, denoted by \iint_S F \cdot n d\sigma, quantifies the net flow of a vector field F through a given surface S.

True, the surface integral of flux, typically denoted by \iint_S F \cdot n d\sigma, is used to quantify the net flow of a vector field F through a specific surface S.

True or False: Divergence (represented as ∇ \cdot F) is a scalar measure indicating the net outward flow of a vector field from an infinitesimal volume, where a positive value signifies a source and a negative value signifies a sink.

True, divergence (represented as ∇ \cdot F) is a scalar measure that indicates the net outward flow (or 'outflow') of a vector field from an infinitesimal volume. A positive value implies a source, while a negative value signifies a sink.

True or False: Curl (represented as ∇ \times F) is a vector that defines both the local axis and magnitude of rotation within a vector field, with its direction determined by the right-hand rule.

True, curl (represented as ∇ \times F) is a vector that describes both the local axis and the magnitude of rotation within a vector field, with its direction determined by the right-hand rule.

True or False: An irrotational field is a vector field where its curl is identically zero everywhere (∇ \times F = 0).

True, an irrotational field is specifically defined as a vector field where its curl is identically zero everywhere (∇ \times F = 0), indicating no local rotation.

True or False: A solenoidal, or incompressible, field is a vector field where its divergence is identically zero everywhere (∇ \cdot F = 0).

True, a solenoidal, or incompressible, field is characterized by its divergence being identically zero everywhere (∇ \cdot F = 0), meaning there are no sources or sinks of flow within the field.

True or False: For any sufficiently smooth scalar function f, the curl of its gradient, ∇ \times (∇f), always equals zero, indicating that all gradient fields are irrotational.

True, a fundamental identity in vector calculus states that for any sufficiently smooth scalar function f, the curl of its gradient, ∇ \times (∇f), always equals zero. This property confirms that all gradient fields are irrotational.

True or False: For any sufficiently smooth vector field G, the divergence of its curl, ∇ \cdot (∇ \times G), always equals zero, which means that fields that are curls of another field are divergence-free.

True, another fundamental identity in vector calculus states that for any sufficiently smooth vector field G, the divergence of its curl, ∇ \cdot (∇ \times G), always equals zero. This implies that any field that can be expressed as the curl of another field must be divergence-free.

True or False: For a 2-D vector field F = P i + Q j defined on a simply-connected domain, F is conservative if and only if the partial derivative of P with respect to y equals the partial derivative of Q with respect to x (\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}).

True, for a 2-D vector field F = P i + Q j defined on a simply-connected domain, a necessary and sufficient condition for F to be conservative is that the cross-partial derivatives are equal: \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} This is a common test for conservativeness in 2D.

True or False: For a conservative force F, the work done by F around any closed path is zero, and the work done along any path depends only on the starting and ending points, not the path itself.

True, a defining characteristic of conservative forces is that the work done by such a force around any closed path is zero. Consequently, the work done along any path depends solely on the starting and ending points, not the specific path taken between them.

True or False: The curl operator exhibits linearity, meaning that for any two vector fields F and G, the curl of their sum is equal to the sum of their curls: ∇ \times (F + G) = ∇ \times F + ∇ \times G.

True, the curl operator is linear, which means it distributes over vector addition. This property is expressed as ∇ \times (F + G) = ∇ \times F + ∇ \times G for any vector fields F and G.

True or False: It is generally valid to apply the curl operator (∇ \times) directly to the scalar result of a dot product between two vector fields (F \cdot G) to form an expression like ∇ \times (F \cdot G).

False, the curl operator (∇ \times) is defined for vector fields, not scalar quantities. The dot product of two vector fields (F \cdot G) results in a scalar field, and thus, applying the curl directly to this scalar field (e.g., ∇ \times (F \cdot G)) is not a valid operation in standard vector calculus.

True or False: For a constant vector field F, the net flux through any closed surface S is always zero, because the integral of the outward normal vector n over a closed surface results in a zero vector.

True, for a constant vector field F and any closed surface S, the net flux through S is indeed zero. This is because the integral of the outward normal vector n over a closed surface, \iintS n d\sigma, always yields the zero vector, making F \cdot (\iintS n d\sigma) = 0.

True or False: The Divergence Theorem (Gauss's Theorem) states that the flux of a vector field F through a closed surface S is equal to the volume integral of the divergence of F over the region W enclosed by S: \iintS F \cdot n d\sigma = \iiintW ∇ \cdot F dV.

True, the Divergence Theorem (also known as Gauss's Theorem) is a fundamental theorem that relates the flux of a vector field F through a closed orientable surface S to the volume integral of the divergence of F over the solid region W enclosed by S. The theorem is precisely stated as \iintS F \cdot n d\sigma = \iiintW ∇ \cdot F dV.

True or False: If the divergence of a vector field F is positive (∇ \cdot F > 0) everywhere within a region W, then the net flux through the boundary surface S of W must be positive, even if F does not strictly point outward at every single point on S.

True, if the divergence of a vector field F is positive (∇ \cdot F > 0) throughout a region W, the Divergence Theorem implies that the net flux out of the boundary surface S of W must be positive. This doesn't mean F points outward at every individual point, but rather that the overall flow is outward.

True or False: If all flow lines of a vector field are straight lines, it inherently means that the curl of the field is zero everywhere (∇ \times F = 0), as no rotational effects are present.

False, while fields with straight flow lines can have zero curl (e.g., a uniform flow like F = c \mathbf{i}), straight flow lines do not guarantee zero curl. For example, the field F = y \mathbf{i} has straight, horizontal flow lines, but its curl is -\mathbf{k} since the magnitude of the field changes perpendicular to the flow lines, indicating local rotation.

True or False: If a vector field's flow is confined entirely within planes parallel to the xy-plane, then any non-zero curl it possesses must be directed parallel to the k vector (i.e., perpendicular to the plane of flow).

True, if a vector field's flow is confined exclusively within planes parallel to the xy-plane (meaning its z-component is zero and has no dependence on z for its x and y components), then any non-zero curl it possesses must be directed perpendicular to these planes, which is parallel to the k vector.

True or False: A vector field G can be expressed as the curl of another vector field H (such that ∇ \times H = G) if and only if G is divergence-free (∇ \cdot G = 0) and its domain is simply-connected.

True, a vector field G can indeed be expressed as the curl of another vector field H (i.e., ∇ \times H = G) if and only if G is divergence-free (∇ \cdot G = 0) and its domain is simply-connected. This condition stems from the identity that the divergence of a curl is always zero, so ∇ \cdot (∇ \times H) = 0.

True or False: A positive total flux of a vector field through a surface implies that the vector field F makes an acute angle with the outward normal vector n at every point on that surface.

False, a positive total flux through a surface only indicates that the net flow is outward. It does not imply that the vector field F makes an acute angle with the outward normal vector n at every point on the surface; there can be regions where F points inward, as long as the outward flow in other regions outweighs it.

True or False: If two vector fields F and G have equal divergences (∇ \cdot F = ∇ \cdot G) throughout a region, then this guarantees that the two vector fields themselves must be identical (F = G).

False, knowing that two vector fields F and G have equal divergences (∇ \cdot F = ∇ \cdot G) throughout a region is not sufficient to guarantee that the two vector fields themselves are identical (F = G). Additional information, such as their curls being equal and specific boundary conditions, is typically required for uniqueness.

True or False: According to the Right-Hand Rule, the direction of the curl of a vector field (∇ \times F) points in the direction of your thumb when your fingers curl in the sense of the field's rotation.

True, the Right-Hand Rule is a convention used to determine the direction of the curl of a vector field. If you curl the fingers of your right hand in the sense of the field's rotation, your thumb will point in the direction of ∇ \times F.

True or False: In vector calculus, the standard positive orientation for closed curves in the xy-plane is counter-clockwise, and traversing the curve in a clockwise direction will reverse the sign of the line integral.

True, in the standard conventions of vector calculus, particularly when working in the xy-plane, a counter-clockwise orientation for closed curves is designated as positive. Consequently, if such a curve is traversed in a clockwise direction, the sign of its line integral will be reversed (multiplied by -1).