Vector Calculus: Gradients, Divergence, and Curl

Module I

• If f(x, y, z) = 3x²y + 2y - 3z, then the gradient ∇f = (6xy, 3x² + 2, -3).
• The gradient of a scalar function φ(x, y, z) is given by i(∂φ/∂x) + j(∂φ/∂y) + k(∂φ/∂z).
• The derivative of the vector function r(t) = t²i + 3tj + 2t²k is r'(t) = 2ti + 3j + 4tk.
• The divergence of a vector function A = 3x²i + 5xy²j + xyz³k is 6x + 10xy + 3xyz².
• If r = xi + yj + zk, then the divergence of r, ∇⋅r = 3.
• A vector field F is solenoidal if its divergence is zero (∇⋅F = 0).

Divergence and Curl of Vector Fields

• Divergence and curl of a vector field are scalar and vector, respectively.
• A vector field F is irrotational if its curl is zero (∇ × F = 0).
• If r = xi + yj + zk, then ∇ × r is 0.
• If A and B are irrotational vectors, then A × B is solenoidal.
• If F is irrotational and F = ∇φ, then φ is called the scalar potential.
• A necessary and sufficient condition for the line integral ∫F⋅dr to vanish for every closed curve C is curl F = 0.
• The directional derivative is grad φ ⋅ n.
• If F = xy²i + 2x²yzj - 3y²zk, then div F is none of these.

Angle Between Surfaces

• The angle between the surfaces x² + y² + z² = 5 and x² + y² + z² - 2x = 5 at (0, 1, 2) is cos⁻¹(√20/√24).
• The value of ∇²(rⁿ) is n(n + 1)rⁿ⁻².
• If φ and ψ are differentiable scalar fields, then ∇ × ∇ψ is solenoidal.
• If A is a constant vector, then curl div A = 0.
• If A = sin(t)i - cos(t)j, then d/dt (A⋅A) = 0.
• Gradient of a scalar function is a vector, and the divergence of a constant is zero.
• Divergence of a vector is a scalar, and curl of a vector is a vector.

Module 2 / Unit V

• The Gauss divergence theorem is given by ∬(P dy dz + Q dz dx + R dx dy) = ∭(∂P/∂x + ∂Q/∂y + ∂R/∂z) dx dy dz.
• The value of ∮(2x - y)dx - yz²dy - y²z dz over the circle x² + y² = 1 corresponding to the surface of the sphere of unit radius is π.
• Using the Gauss divergence theorem to evaluate the surface integral ∬(3xi + 2yj)⋅dS, where S is the sphere given by x² + y² + z² = 9, the value is 180π.
• The Gauss divergence theorem converts a surface to volume integral.
• Green's theorem relates a line and surface integral.
• Using the divergence theorem, the value for the function given by (eᶻ, sin x, y²) is 0.

Gauss Divergence Theorem

• The Gauss divergence theorem states that ∬F⋅nds = ∭(∇⋅F)dV.
• The value of Green's theorem for U = x² and V = y² is 0.
• Applications of Green's theorem are two-dimensional.
• Green's theorem is related to Stoke's theorems mathematically.
• The value of Stoke's theorem for yi + zj + xk is -i - j -k.
• Stoke's theorem converts line integral into surface integral
• In the surface integral ∬ F⋅n̂ dS = ∬ F⋅n̂ dx dz where R is the projection of S on the xz plane.

Green and Stoke's Theorems

• ∫P dx + Q dy + R dz = ∬ (R_y - Q_z) dy dz + (P_z - R_x) dz dx + (Q_x - P_y) dx dy is scalar form of Stoke's theorem.
• The value of ∮4y dx + 7x dy over the circle x² + y² = 4 using Green's Theorem is 12π.
• The value of ∮x dx + y dy + z dz over the circle x² + y² + z² = 16, z = 0 by Stoke's theorem is 0.
• The value of ∭ dV over the sphere x² + y² + z² = r² is (4/3)πr³.
• The value of ∬ V⋅r dV is 3v.

Module 3

• L[k] is k/s.
• L[t] is 1/s².
• L[sin at] is a/(s² + a²).
• L[cos at] is s/(s² + a²).
• L[sinh at] is a/(s² - a²).
• L[cosh at] is s/(s² - a²).
• If L[f(t)] = φ(s), then L[e⁻ᵃᵗf(t)] = φ(s + a).
• L[t sin at] is (2as)/(s² + a²)².
• L⁻¹[1/(s + a)²] is te⁻ᵃᵗ.

Laplace Transforms

• L[t eᵃᵗ sin at] = (s-a)/((s-a)² + a²).
• L⁻¹[s/(s²+4)²] = (1/4)t sin 2t + (1/8)cos 2t.
• L⁻¹[1/(s(s²+1))] is 1 - cos t.
• L⁻¹[(s - a)/((s - a)² + b²)] is eᵃᵗ cos bt.
• L⁻¹[1/√s] is tⁿ⁻¹/(n-1)!.
• L[f'(t)] = sL[f(t)] - f(0).
• L[3e⁵ᵗ + 5 cos t] = (3/(s - 5)) + (5s/(s² + 1)).
• A function f(t) is of exponential order if lim(t→∞) e⁻ˢᵗf(t) = 0.

Inverse Laplace Transforms

• L⁻¹[s/(s² – a²)] is cosh(at).
• If f(t) is a periodic function with period P, then the Laplace transform of f(t) is (1/(1-e^(-Ps))) ∫₀^P e^(-st)f(t) dt.
• What is the initial value theorem in Laplace transforms? : lim(s→∞) sF(s) = f(0)
• What is the final value theorem in Laplace transforms? lim(s→0) sF(s) = f(∞)

Module 4

• L⁻¹[1/(s+a)] is e⁻ᵃᵗ.
• L⁻¹[1/(s(s²+1))] is 1 - cos t.
• L⁻¹[(s - a)/((s - a)² + b²)] is eᵃᵗ cos bt.
• L⁻¹[1/sⁿ⁺¹] is tⁿ/n!.

Inverse Laplace Transform Validations

• L⁻¹[s/(s² - a²)] is cosh at.
• L⁻¹[1/(s + a)] is valid for s > -a.
• The inverse Laplace transform of 1/s is 1.
• Partial Fraction Method is used to find the inverse Laplace transform of a rational function.
• Convolution Theorem is used for in Laplace transforms to find the inverse Laplace transform of a product of two functions.

Properties of Inverse Laplace Transforms

• The property of inverse Laplace transforms that states L⁻¹{F(s)G(s)} = f(t) * g(t) is Convolution Property.
• The inverse Laplace transform of 1/(s-a) is eᵃᵗ.
• The inverse Laplace transform of 1/s² is t.
• The property of inverse Laplace transforms that states L⁻¹{F(s-a)} = eᵃᵗ L⁻¹{F(s)} is Shifting Property.
• The inverse Laplace transform of s/(s²+1) is cos(t).
• The inverse Laplace transform of 1/(s² - 4) is (1/2)sinh(2t). L⁻¹{dF(s)/ds} = -tf(t) is Differentiation Property.

Module 5

• inverse Laplace transform of s/(s²+1) is cos(t).
• If f(t) is a periodic function with period P, then Laplace transform of f(t) is (1/(1-e^(-Ps))) ∫₀^P e^(-st)f(t) dt
• The Laplace transform of the unit step function u(t) is 1/s.
• Laplace transform of the unit impulse function δ(t) is 1.
• Laplace transform of the Dirac delta function ō(t-a) is e^(-as). Laplace transform of the function f(t) = e^(at)u(t) is 1/(s-a).
• Laplace transform of the function f(t) = t^n u(t) is n!/s^(n+1).