Essential Calculus
2nd Edition
ISBN: 9781285209067
Author: Stewart
Publisher: Cengage
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Let r be a di¡erentiable vector function of t. Show that if r # (dr/dt) = 0 for all t, then r is constant.
Find ∂ w/∂ x if w = x2 + y2 + z2 and z = x2 + y2.
Express
∂w∂u
and
∂w∂v
as functions of u and v.
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- 2. Let U be a vector function of position in R³ with continuous second partial deriva- tives. We write (U₁, U2, U3) for the components of U. (a) Show that V (U • U) – Ü × (▼ × Ü) = (U · ▼) Ū, (V where ((UV) U), = U₁ (b) We define the vector function of position, by setting = V × . If the condition V U = 0 holds true, show that ▼ × ((Ū · V) Ű) = (Ũ · V) Ñ - (Ñ· ▼) ū. (Hint: The representation of (UV) U from the first part might be useful.)arrow_forwardProve that the directiongl derivative of fla,y,z) =zysinz direction v=°i+2j+2k IS (1,2,5)in the is 4 Z atarrow_forward2. Sketch the vector field F(x, y, z) = xi + yj+ zk.arrow_forward
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