Let T be the transformation in Quick Check Exercise 1. (a) The Jacobian ∂ x , y / ∂ u , υ of T is ______ . (b) Let R be the region in Quick Check Exercise 1(a). Fill in the missing integrand and limits of integration for the change of variables given by T . ∬ R e x + 2 y d A = ∫ ∫ _____ d u d υ
Let T be the transformation in Quick Check Exercise 1. (a) The Jacobian ∂ x , y / ∂ u , υ of T is ______ . (b) Let R be the region in Quick Check Exercise 1(a). Fill in the missing integrand and limits of integration for the change of variables given by T . ∬ R e x + 2 y d A = ∫ ∫ _____ d u d υ
Let T be the transformation in Quick Check Exercise 1.
(a) The Jacobian
∂
x
,
y
/
∂
u
,
υ
of T is
______
.
(b) Let R be the region in Quick Check Exercise 1(a). Fill in the missing integrand and limits of integration for the change of variables given by T.
∬
R
e
x
+
2
y
d
A
=
∫
∫
_____
d
u
d
υ
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
a) Find the directional derivative of f(x, y, z) = x²yz + 4xz² at (1, -2, 1) in the direction 2i - j - 2k.
Consider the function f : R³ → R defined by
f(x, y, z) = tanh(5x² + 2y² - 3z²)
and the point P: (x, y, z) = (1, 1, 1).
(a) In what direction is f most rapidly increasing at the point P?
(b) Calculate the directional derivative of f in the direction of u = (2, -1, 2) at the point P.
(c) Determine an expression for the linear approximation f of f at point P.
(d) Use the linear approximation to approximate the value of f(-).
Verify Stokes theorem for the function f=-yi+xj in the square with corners at (0,0,0) and (1,1,0).
University Calculus: Early Transcendentals (3rd Edition)
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