mech analysis 1 review

a scalar field is best described as

• A) A vector assigned to every point in space

• B) A single number assigned to every point in space

• C) A curve parameterized by t

• D) A surface parameterized by u and v

one number (magnitude only) at each point

which of the following is a vector field?

• A) Temperature in a room

• B) Pressure in a fluid

• C) Velocity of flow in a fluid

• D) Electric potential

C. velocity of flow in a fluid. temperature, pressure and electric potential are scalar fields

the gradient f of a scalar field f points in the direction of:

• A) Zero change

• B) Maximum rate of increase of f

• C) Maximum rate of decrease of f

• D) The nearest contour line

B. the gradient points toward maximum increase, with magnitude equal to the maximum rate of change.

at any point, the gradient vector is_____to the contours curve(2D) or contour surface (3D) of f through that point.

• A) Parallel

• B) Perpendicular

• C) At 45°

• D) Tangent

B. The gradient is perpendicular to contours.

the gradient of f(x,y,z) = x²y+z is:

• A) [2xy, x², 1]

• B) [x², 2xy, 1]

• C) [2x, 1, 1]

• D) [2xy, x², z]

A. ∂f/∂x = 2xy, ∂f/∂y = x², ∂f/∂z = 1.

The directional derivative of f at point P in the direction of unit vector v̂ is:

• A) ∇f × v̂

• B) ∇f · v̂

• C) ∇·f

• D) |∇f| + |v̂|

B. D_v̂ f = ∇f · v̂ = |∇f| cos θ.

If the angle θ between ∇f and the unit vector v̂ is π/2, the directional derivative is:

• A) |∇f|

• B) −|∇f|

• C) 0

• D) 1

C. cos(π/2) = 0, so there is no change (movement along a contour).

If θ = π (180°) between ∇f and v̂, the directional derivative equals:

• A) |∇f|

• B) 0

• C) −|∇f|

• D) It is undefined

C. cos π = −1, the direction of steepest decrease.

For f(x, y) = x² + y², the directional derivative at (1, 1) in the direction of the unit vector [1, 0] is:

• A) 0

• B) 1

• C) 2

• D) 2√2

C. ∇f = [2x, 2y] = [2, 2] at (1, 1); [2, 2]·[1, 0] = 2.

Divergence takes a ______ and produces a ______.

• A) scalar field; vector field

• B) vector field; scalar field

• C) vector field; vector field

• D) scalar field; scalar field

B. Divergence maps a vector field to a scalar field.

For F(x, y) = [x, y], the divergence ∇·F is:

• A) 0

• B) 1

• C) 2

• D) −2

C. ∂x/∂x + ∂y/∂y = 1 + 1 = 2.

For F(x, y) = [−x, −y], the field vectors are:

• A) Expanding away from the origin (source)

• B) Converging toward the origin (sink)

• C) Neither expanding nor contracting

• D) Rotating counter-clockwise

B. ∇·F = −2 < 0: vectors converge (a sink).

The divergence of F(x, y) = [2x − y, y²] is:

• A) 2 − 2y

• B) 2 + 2y

• C) 2x + 2y

• D) 0

B. ∂(2x − y)/∂x = 2 and ∂(y²)/∂y = 2y.

The Laplacian ∇²f in two dimensions is:

• A) ∂f/∂x + ∂f/∂y

• B) ∂²f/∂x² + ∂²f/∂y²

• C) ∂²f/∂x∂y

• D) [∂f/∂x, ∂f/∂y]

B. The sum of the second partials.

The Laplacian is defined as the:

• A) Curl of the gradient

• B) Gradient of the divergence

• C) Divergence of the gradient

• D) Curl of the curl

C. ∇·∇f = ∇²f.

The 2D curl of F = [f₁, f₂] is:

• A) ∂f₁/∂x + ∂f₂/∂y

• B) ∂f₂/∂x − ∂f₁/∂y

• C) ∂f₁/∂y − ∂f₂/∂x

• D) ∂f₂/∂x + ∂f₁/∂y

B. curl = ∂f₂/∂x − ∂f₁/∂y.

A positive 2D curl at a point indicates:

• A) Clockwise rotation

• B) Counter-clockwise rotation

• C) No rotation

• D) Expansion of the field

B. Positive means counter-clockwise; negative means clockwise; zero means irrotational.

For F = [y³ − 9y, x³ − 9x], the curl at the point (0, 2) is:

• A) 12

• B) 0

• C) −12

• D) 3

C. The curl is 3x² − 3y². At (0, 2): 0 − 12 = −12 (clockwise).

The 3D curl of F = [y³ − 9y, x³ − 9x, 0] is:

• A) [3x² − 3y², 0, 0]

• B) [0, 3x² − 3y², 0]

• C) [0, 0, 3x² − 3y²]

• D) [0, 0, 0]

C. The z-component is ∂f₂/∂x − ∂f₁/∂y = (3x² − 9) − (3y² − 9) = 3x² − 3y². The other components vanish.

In three dimensions, the direction of the curl vector gives:

• A) The direction of maximum increase

• B) The axis of rotation (right-hand rule)

• C) The direction of expansion

• D) The tangent to the streamline

B. The curl vector's direction is the rotation axis (right-hand rule); its magnitude is the intensity of rotation.

The standard parameterization of the straight line from point A to point B is:

• A) r(t) = r_B + t(r_A)

• B) r(t) = r_A + t(r_B − r_A), 0 ≤ t ≤ 1

• C) r(t) = t(r_A + r_B)

• D) r(t) = r_A − t(r_B − r_A)

B. r(t) = r_A + t(r_B − r_A) runs from A at t = 0 to B at t = 1.

Which vector function parameterizes a helix of radius r?

• A) [r cos t, r sin t, 0]

• B) [r cos t, r sin t, t]

• C) [t cos t, t sin t, r]

• D) [r cos t, t, r sin t]

B. The circular motion in x, y plus linear growth in z.

The work done by a vector field F along a curve C parameterized by r(t), a ≤ t ≤ b, is:

• A) ∫ₐᵇ F(r(t)) × r′(t) dt

• B) ∫ₐᵇ F(r(t)) · r′(t) dt

• C) ∫ₐᵇ |F(r(t))| dt

• D) ∫ₐᵇ F(r(t)) · r(t) dt

B. W = ∫ F · dr = ∫ F(r(t)) · r′(t) dt (a dot product, not a cross product).

The unit tangent vector T̂ of a curve r(t) is:

• A) r(t)/|r(t)|

• B) r′(t)

• C) r′(t)/|r′(t)|

• D) r″(t)/|r″(t)|

C. Normalize the velocity vector.

The arc-length element ds equals:

• A) |r(t)| dt

• B) r′(t) dt

• C) |r′(t)| dt

• D) |r″(t)| dt

C. ds/dt = |dr/dt|, i.e. speed = magnitude of velocity.

A surface is parameterized by r(u, v). The vector normal to the tangent plane at r(u₀, v₀) is:

• A) r_u · r_v

• B) r_u + r_v

• C) r_u × r_v

• D) r_u − r_v

C. r_u and r_v are both tangent to the surface, so their cross product is normal.

The surface area of a parameterized surface S is:

• A) ∬ r_u · r_v dA

• B) ∬ |r_u × r_v| dA

• C) ∬ (r_u × r_v) dA

• D) ∬ |r_u||r_v| dA

B. |r_u × r_v| is the area of the small parallelogram spanned by r_u and r_v.

The cylinder wall of radius a and height h is parameterized by r(u, v) = [a cos u, a sin u, v]. The correct domain is:

• A) 0 ≤ u ≤ π, 0 ≤ v ≤ h

• B) 0 ≤ u ≤ 2π, 0 ≤ v ≤ h

• C) 0 ≤ u ≤ h, 0 ≤ v ≤ 2π

• D) 0 ≤ u ≤ 2π, 0 ≤ v ≤ 2π

B. u is the angle (full rotation 0 to 2π), v is the height 0 to h.

For the sphere r(u, v) = [R cos u cos v, R sin u cos v, R sin v] with 0 ≤ u ≤ 2π and −π/2 ≤ v ≤ π/2, the magnitude |r_u × r_v| is:

• A) R

• B) R²

• C) R² cos v

• D) R² sin v

C. r_u × r_v = [R² cos u cos²v, R² sin u cos²v, R² cos v sin v], whose magnitude is R² cos v (non-negative on the domain).

Using the parameterization above, the surface area of the sphere is:

• A) 2πR²

• B) 4πR²

• C) 4πR³/3

• D) πR²

B. ∫₀^{2π} ∫_{−π/2}^{π/2} R² cos v dv du = 2π · 2R² = 4πR².

A scalar field has a magnitude and a direction at every point. True or False

False. A scalar field has magnitude only, with no direction. A vector field has both.

32. The gradient of a scalar field is a vector field. true or false

True. ∇f assigns a vector to each point.

33. The directional derivative is largest when the unit vector points in the same direction as the gradient. true or false

True. When θ = 0, D = |∇f|, the maximum.

34. If the directional derivative of f in the direction v̂ is zero, then v̂ is parallel to ∇f. true or false

False. A zero directional derivative means v̂ is perpendicular to ∇f (θ = 90°).

35. The divergence of a vector field is itself a vector. true or false

False. Divergence is a scalar.

36. A negative divergence means more field vectors enter a small region than leave it. true or false

True. That is the definition of a sink or converging field.

37. The field F = [y, x] has zero divergence. true or false

True. ∂y/∂x + ∂x/∂y = 0 + 0 = 0.

38. The field F = [y, x] has non-zero 2D curl. true or false

False. curl = ∂x/∂x − ∂y/∂y = 1 − 1 = 0. It is irrotational.

39. A field with zero divergence everywhere must be the zero vector field. true or false

False. Zero divergence only means no net expansion or contraction (e.g. F = [y, x] is nonzero but divergence-free).

40. The Laplacian of f is the divergence of the gradient of f. true or false

True. ∇·(∇f) = ∇²f.

41. For F = [f₁(x, y), f₂(x, y), 0], the 3D curl has only a z-component. true or false

True. Both the first two curl components involve ∂/∂z or f₃, which are zero, leaving [0, 0, ∂f₂/∂x − ∂f₁/∂y].

42. Reversing the direction in which a curve is traversed does not change the sign of the work done by a vector field. true or false

False. Reversing direction flips r′(t), so the work changes sign.

43. r′(t) is the velocity vector of a particle at position r(t). true or false

True. The derivative of position with respect to time is velocity.

44. If the vector field is perpendicular to the path at every point, the work done is zero. true or false

True. F · T̂ = |F| cos 90° = 0 at every point, so the integrand is zero.

45. A surface in space needs only one parameter, just like a curve. true or false

False. Curves need one parameter; surfaces need two (u and v).

46. The cross product r_u × r_v is tangent to the surface at the point r(u₀, v₀). true or false

False. It is normal to the surface (perpendicular to the tangent plane).

47. A full sphere is not the graph of a single function z = f(x, y), but each hemisphere is. true or false

True. The upper and lower hemispheres are z = ±√(R² − x² − y²), but the full sphere fails the vertical line test.