Multivariable Calculus Formulas

Question 1

Equation of a tangent plane to z = f(a,b)

Answer 1

z = f(a,b) + df/dx(a,b).(x - a) + df/dy(a,b).(y - b)

Question 2

Classification of critical points in the (x,y) plane

Answer 2

• |H| > 0, fxx < 0 -> local maximum
• |H| > 0, fxx > 0 -> local minimum
• |H| < 0 -> saddlepoint
• |H| = 0 -> unknown

Question 3

Change of variables:
dxdy -> dudv

Answer 3

dxdy = |d(x,y)/d(u,v)| dudv

Question 4

Change of variables:
dxdy -> drd(theta)

Answer 4

dxdy = r * drd(theta)

Question 5

Change of variables:
dxdydz -> drd(theta)d(phi)

Answer 5

dxdydz = r^2*sin(theta) * drd(theta)d(phi)

Question 6

Change of variables:
(x, y) -> (r, theta)

Answer 6

x = r * cos(theta)
y = r * sin(theta)

Question 7

Change of variables:
(x, y, z) -> (r, theta, phi)

Answer 7

x = r * sin(theta) * cos(phi)
y = r * sin(theta) * sin(phi)
z = r * cos(theta)

Question 8

Canonical form of a circle

Answer 8

x^2 + y^2 = r^2

Question 9

Canonical form of an elipse

Answer 9

x^2/a^2 + y^2/b^2 = r^2

Question 10

Canonical form of a hyperbolae

Answer 10

x^2/a^2 - y^2/b^2 = r^2

Question 11

How do you determine if a function from (x, y) -> (u, v) is invertible in (u, v)

Answer 11

Invertibility theorem:
d(x, y) / d(u, v) != 0

Question 12

Line integral equation equal to the area of an enclosed (simple-connected) domain

Answer 12

Question 13

Direction and magnitude of steepest ascent

Answer 13

Direction of delta.f = (df/dx.i + df/dy.j)
Magnitude of delta.f

Question 14

Taylor expansion of f(x, y) about (a, b)

Answer 14

Question 15

Directional derivative

Answer 15

D(u) = (df/dx*i + df/dy*j).(ai + bj)/|ai + bj|

Question 16

Cos double angle formula

Answer 16

cos(2*theta) = cos^2(theta) - sin^2(theta)

Question 17

Sin double angle formula

Answer 17

sin(2*theta) = 2*cos(theta)*sin(theta)

Question 18

Direction of steepest ascent

Answer 18

v = df/dx*i + df/dy*j

Question 19

Directions of constant height

Answer 19

Two directions orthogonal to v, the direction of steepest ascent (+,-)

Question 20

Path of steepest ascent

Answer 20

dy/dx = df/dy / df/dx

Question 21

J(u,v)

Answer 21

d(x,y) / d(u,v)

Question 22

Green’s theorem

Answer 22

Question 23

Lagrange multiplier equations

Answer 23

d(f,g)/d(x,y) = 0
g(x,y) = 0 (constraint)