代做21-259: Calculus in Three Dimensions Lecture #5 Spring 2025代做留学生SQL语言程序
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Lecture #5
Spring 2025
Vector Functions and Space Curves
Definition: A vector function r (t) : R→R n is a function whose domain is a set of real numbers and whose range is a set of vectors.
For n = 3, r (t) =
Example 1. Find the domain of the vector function r (t) = <3 ,ln(3− t), √t>.
If r (t) =
provided the limits of the component functions exist.
Example 2. Find
A vector function r (t) is continuous at t = a if r (t) = r (a).
Definition: Let f , g , and h be continuous functions on an interval I. Then the set C of all points (x, y, z) in space, where
x = f (t), y = g (t), z = h(t),
is called a space curve. The equations above are the parametric equations of C, and t is called a parameter. Any continuous vector function r (t) defines a space curve.
Example 3. What are the parametric equations of a circle of radius a in the x y-plane, centered at the origin?
Example 4. What is the curve given by r (t) = cos tı +sint ȷ + tk?
Example 5. Find a vector function that represents the curve of intersection of the cylinder x2 + y2 = 1 and the plane y + z = 2.
Example 6. Find a vector function that represents the curve of intersection of the paraboloid z = 4x2 + y2 and the parabolic cylinder y = x 2 .
Calculus of Vector Functions
Definition: The derivative r′ of a vector function r is given by
provided the limit exists.
For any value of t, the vector r′ (t) is the tangent vector to the curve defined by r, provided that r′ (t) exists and r′ (t) ≠ 0. The vector
is the unit tangent vector to r (t).
Theorem: If r (t) =
r′ (t) = <f′ (t), g′ (t),h′ (t)> = f′ (t)ı + g′ (t) ȷ +h′ (t)k.
Example 7. For the vector function r (t) =
Example 8. Find parametric equations for the tangent line to the curve x = lnt, y = 2√t, z = t 2 at the point (0,2,1).
Theorem: Suppose u and v are differentiable vector functions, c is a scalar, and f is a real-valued function. Then
1. dt/d (u(t)+ v(t)) = u ′ (t)+ v ′ (t)
2. dt/d (cu(t)) = cu′ (t)
3. dt/d (f (t)u(t)) = f (t)u′ (t)+ f′ (t)u(t)
4. dt/d (u(t)· v(t)) = u(t)· v′ (t)+u′ (t)· v(t)
5. dt/d (u(t)× v(t)) = u(t)× v′ (t)+u′ (t)× v(t)
6. dt/d (u(f (t))) = f′ (t)u′ (f (t))
Example 9. If r (t) ≠ 0, show that
Example 10. Show that if |r (t)| = c (where c is a nonzero constant) then r′ (t) is orthogonal to r (t) for all t.
Example 11. Show that if r is a vector function such that r′′ exists, then
Definition: The definite integral of a continuous vector function r (t) =
Example 12. Evaluate