Final Practice

Question 1

What does the gradient vector at a point represent in a multivariable function?

Answer 1

It points in the direction of the steepest ascent (maximum rate of increase) and its magnitude is the rate of that increase.

Question 2

How do you find the direction of steepest ascent for a function f(x,y) at a point?

Answer 2

Compute the gradient ∇f at the point; it is the vector direction.

Question 3

What vectors represent directions of no change in height (level curves) for a function f(x,y)?

Answer 3

Vectors perpendicular to the gradient ∇f, such as <a, b> perpendicular to <-b, a> or <b, -a>.

Question 4

How do you compute the directional derivative in the direction of a vector v?

Answer 4

Normalize v to a unit vector u = v / ||v||, then compute ∇f · u.

Question 5

What does the sign of the directional derivative indicate?

Answer 5

Positive: uphill (increasing); negative: downhill (decreasing); zero: no change.

Question 6

How do you determine if a path is uphill or downhill based on directional derivative?

Answer 6

If >0, uphill; if <0, downhill.

Question 7

What is a conservative vector field?

Answer 7

A field F where the line integral is path-independent, equivalent to F = ∇f for some potential function f.

Question 8

How do you find a potential function for a conservative vector field <P, Q, R>?

Answer 8

Integrate P wrt x (or similar), then add terms from Q and R, ensuring consistency; add constant if needed.

Question 9

How do you compute a line integral over a curve C for a conservative field?

Answer 9

Use the fundamental theorem: f(end point) - f(start point), where F = ∇f.

Question 10

When should you use the easy way vs. hard way for line integrals?

Answer 10

Easy: If conservative, use potential and endpoints; hard: Parametrize and integrate directly.

Question 11

What theorem relates line integrals around closed curves to double integrals?

Answer 11

Green’s Theorem: ∫_C P dx + Q dy = ∬_R (∂Q/∂x - ∂P/∂y) dA.

Question 12

How do you compute outward flux through a closed curve using a theorem?

Answer 12

Use Green’s Theorem: ∫_C -Q dx + P dy = ∬_R (∂P/∂x + ∂Q/∂y) dA.

Question 13

What theorem relates flux through a closed surface to a triple integral?

Answer 13

Divergence Theorem: ∬_S F · dS = ∭_V ∇·F dV.

Question 14

How do you apply the Divergence Theorem for outward flux?

Answer 14

Compute ∇·F, then triple integrate over the enclosed volume; easy if ∇·F is constant.

Question 15

What theorem relates line integrals over a boundary to surface integrals?

Answer 15

Stokes’ Theorem: ∫_C F · dr = ∬_S (∇×F) · dS.

Question 16

How do you compute a line integral over a triangular path oriented counter-clockwise?

Answer 16

Use Stokes’ Theorem if applicable: Project to a surface and compute curl integral.

Question 17

How do you compute the surface area of a graph z = g(x,y) over a region R?

Answer 17

∬_R √(1 + (∂g/∂x)^2 + (∂g/∂y)^2) dA.

Question 18

For surface area over a disk, why switch to polar coordinates?

Answer 18

Simplifies the integral due to radial symmetry; dA = r dr dθ.

Question 19

What is the method for finding power series solutions to an ODE about x=0?

Answer 19

Assume y = Σ a_n x^n, compute y’ and y’’, plug in, shift indices, solve for coefficients.

Question 20

How do you find the first few terms (e.g., up to x^4) in a power series solution?

Answer 20

Set up recurrence from equated coefficients, solve starting from a0 and a1.

Question 21

What is a recurrence relation in power series solutions?

Answer 21

A formula relating a_{n+2} (or similar) to previous a_k terms.

Question 22

For a second-order ODE, how many arbitrary constants in the series solution?

Answer 22

Two, typically a0 and a1, with higher terms in terms of them.

Question 23

When shifting indices in series, what ensures alignment?

Answer 23

Adjust summation limits so powers of x match (e.g., let k = n-2).

Question 24

How do you handle ODEs like y’’ + x y’ - y = 0 with series?

Answer 24

Multiply terms by x where needed, then equate coefficients after plugging in series.

Question 25

What is the Frobenius method used for?

Answer 25

Series solutions about singular points of ODEs, assuming y = Σ c_n x^{n+r}.

Question 26

What are indicial roots in Frobenius method?

Answer 26

Roots r of the indicial equation from lowest power terms after plugging in.

Question 27

How do you find the indicial equation for a singular point x=0?

Answer 27

Plug y = x^r Σ c_n x^n into ODE, equate coefficients of x^{r-2} or similar to zero.

Question 28

What is the general recurrence relation in Frobenius?

Answer 28

Formula for c_{n+1} or c_n in terms of previous c_k, involving r and n.

Question 29

For a regular singular point, what form does the ODE take?

Answer 29

x^2 y'' + x p(x) y' + q(x) y = 0, with p and q analytic.

Question 30

If indicial roots differ by integer, what might happen?

Answer 30

One solution may involve log terms, but basic case solves recurrence for each r.

Question 31

What is the Fourier series for a function on [-L, L]?

Answer 31

(a0/2) + Σ [a_n cos(nπx/L) + b_n sin(nπx/L)].

Question 32

How do you compute coefficients for Fourier series of a piecewise function?

Answer 32

a0 = (1/L) ∫ f(x) dx; a_n = (1/L) ∫ f(x) cos(nπx/L) dx; b_n similar with sin.

Question 33

For an odd function, what simplifies in Fourier series?

Answer 33

a_n = 0 (cosine terms vanish); only sine series.

Question 34

How do you integrate for b_n in a piecewise constant function?

Answer 34

Split integral over pieces, compute sin or cos at bounds.

Question 35

Why is the interval (e.g., -π to π) important in Fourier?

Answer 35

Determines the fundamental period and frequencies nπx/L.

Question 36

What is the heat equation form?

Answer 36

u_t = k u_xx, where k is thermal diffusivity.

Question 37

For heat equation with u(0,t)=u(L,t)=0, what solution method?

Answer 37

Separation of variables: u(x,t) = Σ b_n sin(nπx/L) e^{-k (nπ/L)^2 t}.

Question 38

How do you find coefficients in heat equation solution?

Answer 38

From initial condition: b_n = (2/L) ∫ u(x,0) sin(nπx/L) dx.

Question 39

For piecewise initial temperature, how to compute series?

Answer 39

Use Fourier sine series for the initial function.

Question 40

What is the recursive property of the Gamma function?

Answer 40

Γ(x+1) = x Γ(x), proved by integration by parts on ∫ t^{x} e^{-t} dt.