1. **Sketch, by hand, the following vector fields by sketching a few representative arrows:** (a) \(\vec{F} = \langle 0, 1 \rangle\) (b) \(\vec{F} = \langle 0, x \rangle\) (c) \(\vec{F} = \left\langle \frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2 + y^2}} \right\rangle\) (d) \(\vec{F} = \langle y, -x \rangle\)

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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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4. (a) (b) (c) Consider f(x, y, z) = 2²ev - 2xy. Find the equation of the level surface through the point (0, 1, -1). Find a vector normal to this level surface at the point (0, 1,-1). Is the function increasing or decreasing in the direction of the vector you found in part (b)?
16. If n is a unit vector, then în · Vf is called the directional derivative of f in the direction n. Find the directional derivative of ƒ (x,y) = x²yz in the direction 4 î - 3 k at a point (1,-1, 1).
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Let the vector C represents the coefficients of a polynomial pt=c1+c2t+c3t2+c4t3. Express the conditions p0=1, p'0=2, p1=1, p'1=0 in the form AC=b.
3. Let i, j, k denote the unit vectors along the three coordinate axes. Let v(t) = ti+ sintj + costk and w(t) = 3ti + 2k. (a) Compute (v.w)' (t) directly and check your answer by using the product rule (for the dot product, stated in the class). Note that the dot above, is the dot product of vectors and denotes the derivative. (b) Compute (v x w)'(t) directly and check your answer by using the product rule (for the cross product, again stated in the class). Note that the x above, is the cross product of vectors.
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9.Vector analysis (calculus 4)
Suppose f(x, y) (a) ▼ f(x, y) = (b) ▼ f(-1.1, 6) = = =√√tan(x) + y and u is the unit vector in the direction of (-1, 1). Then, = (c) fu (-1.1, 6) = Du f(−1.1, 6) = =

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1. Sketch, by hand, the following vector fields by sketching a few representative arrows: (a) F = (0, 1) (b) F = (0, x) (2) P-(√²+P' F= (d) F = (y, - x) Y √x² + y²¹ √√√x² + y² 2 /