(a) The surface S is defined by x + y² + z³ = 3. Find a vector normal to S at the point A with position vector a = (1,1,1). Hence, find the equation of the plane tangent to S at A. (b) The curve C is defined by the section of surface S on the (x, y)-plane with x > 0. Sketch C and evaluate the line integral =√₁² where ds is an element of arc length. = I = (c) Recall that the divergence of a vector field F expressed in a spherical polar basis is given, in terms of the radial, polar and azimuthal components Fr, Fe and Fo respectively, by V.F A fluid flows on the surface of a unit sphere in the direction of increasing azimuthal angle. The magnitude u of its velocity u is given by |u| = (1+ sin o) sin(20). Evaluate the divergence of the fluid's velocity field on the spherical surface. 10 r² Ər √4-x+3y² ds (r² Fr) + 1 Ә r sin 020 (sin0 Fe) + OF 1 r sin do
(a) The surface S is defined by x + y² + z³ = 3. Find a vector normal to S at the point A with position vector a = (1,1,1). Hence, find the equation of the plane tangent to S at A. (b) The curve C is defined by the section of surface S on the (x, y)-plane with x > 0. Sketch C and evaluate the line integral =√₁² where ds is an element of arc length. = I = (c) Recall that the divergence of a vector field F expressed in a spherical polar basis is given, in terms of the radial, polar and azimuthal components Fr, Fe and Fo respectively, by V.F A fluid flows on the surface of a unit sphere in the direction of increasing azimuthal angle. The magnitude u of its velocity u is given by |u| = (1+ sin o) sin(20). Evaluate the divergence of the fluid's velocity field on the spherical surface. 10 r² Ər √4-x+3y² ds (r² Fr) + 1 Ә r sin 020 (sin0 Fe) + OF 1 r sin do
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. (a) The surface S is defined by x + y² + z³ = 3. Find a vector normal to S at the
point A with position vector a = (1,1,1). Hence, find the equation of the plane
tangent to S at A.
(b) The curve C is defined by the section of surface S on the (x, y)-plane with x ≥ 0.
Sketch C and evaluate the line integral
Jo
√√4 = x + 3y² ds
where ds is an element of arc length.
(c) Recall that the divergence of a vector field F expressed in a spherical polar basis
is given, in terms of the radial, polar and azimuthal components F, Fe and Fo
respectively, by
V.F
=
I =
1 ə
r² Ər
(r² F₁) +
1 Ә
r sin 020
(sin0 Fe) +
1 OF
r sin do
A fluid flows on the surface of a unit sphere in the direction of increasing azimuthal
angle . The magnitude u of its velocity u is given by |u| (1 + sin o) sin(20).
Evaluate the divergence of the fluid's velocity field on the spherical surface.
=
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