1. Consider x=h(y,z) as a parametrized surface in the natural
way. Write the equation of the...
1. Consider x=h(y,z) as a parametrized surface in the natural
way. Write the equation of the tangent plane to
the surface at the point (5,3,−4) given that ∂h/∂y(3,−4)=1 and
∂h/∂z(3,−4)=0.
2. Find the equation of the tangent plane to the surface
z=0y^2−9x^2 at the point (3,−1,−81). z=?
Evaluate the surface integral.
S
(x + y + z) dS, S is the parallelogram with...
Evaluate the surface integral.
S
(x + y + z) dS, S is the parallelogram with parametric
equations
x = u + v,
y = u − v,
z = 1 + 2u + v,
0 ≤ u ≤ 7,
0 ≤ v ≤ 4.
Write down the parametrized surfaces as level surfaces
{f(x,y,z)=0}.
x=ucosv, y=usinv, z=u, 0 <= u <=...
Write down the parametrized surfaces as level surfaces
{f(x,y,z)=0}.
x=ucosv, y=usinv, z=u, 0 <= u <= 2, 0 <= v <=
2pi
x = 2cosu*cosv, y = 2cosu*sinv, z = 2sinu, 0 <= u <= 2pi,
0 <= v <= pi
if y=uv, where u and v are functions of x, show that the nth
derivative of...
if y=uv, where u and v are functions of x, show that the nth
derivative of y with respect to x is given by
(also known as Leibniz Rule)
Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v;...
Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...
Consider the surface S in R3 defined implicitly by x**2 y =
4ze**(x+y) − 35 ....
Consider the surface S in R3 defined implicitly by x**2 y =
4ze**(x+y) − 35 .
(a) Find the equations of the implicit partial derivatives ∂z ∂x
and ∂z ∂y in terms of x, y, z. (b) Find equations of the tangent
plane and the norma line to the surface S at the point (3, −3,
2)
Use the Chain Rule to evaluate the partial derivative
∂f∂u and ∂f∂u at (u, v)=(−1, −1),...
Use the Chain Rule to evaluate the partial derivative
∂f∂u and ∂f∂u at (u, v)=(−1, −1), where
f(x, y, z)=x10+yz16,
x=u2+v, y=u+v2, z=uv.
(Give your answer as a whole or exact number.)
∂f∂u=
∂f∂v=
Find the equation of the tangent plane (in terms of x, y and z)
to the...
Find the equation of the tangent plane (in terms of x, y and z)
to the surface given by x = u, y = v and z = uv at the point (3, 2,
6).
Two functions, u(x,y) and v(x,y), are said to verify the
Cauchy-Riemann
differentiation equations if they satisfy...
Two functions, u(x,y) and v(x,y), are said to verify the
Cauchy-Riemann
differentiation equations if they satisfy the following
equations ∂u\dx=∂v/dy and ∂u/dy=−(∂v/dx)
a. Verify that the Cauchy-Riemann differentiation equations can
be written in the polar coordinate form as
∂u/dr=1/dr ∂v/dθ and ∂v/dr =−1/r ∂u/∂θ
b. Show that the following functions satisfy the Cauchy-Riemann
differen- tiation equations
u=ln sqrt(x^(2)+y^(2)) and v= arctan y/x.
Verify the Caucy-riemann equations for the functions u(x,y),
v(x,y) defined in the given domain
u(x,y)=x³-3xy², v(x,y)=3x²y-y³,...
Verify the Caucy-riemann equations for the functions u(x,y),
v(x,y) defined in the given domain
u(x,y)=x³-3xy², v(x,y)=3x²y-y³, (x,y)ɛR
u(x,y)=sinxcosy,v(x,y)=cosxsiny (x,y)ɛR
u(x,y)=x/(x²+y²), v(x,y)=-y/(x²+y²),(x²+y²), (
x²+y²)≠0
u(x,y)=1/2 log(x²+y²), v(x,y)=sin¯¹(y/√¯x²+y²), ( x˃0 )
In each case,state a complex functions whose real and imaginary
parts are u(x,y) and v(x,y)