Term
| What is a vector function? |
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Definition
| any function that is defined by its effect on the unit vectors i, j, and k |
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Term
| How do you find the derivative and the integral of a vector function? |
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Definition
| by integrating or differentiating by its components separately |
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Term
| What is the connection between vector function and space curves? |
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Definition
| space curves are determined by vector functions |
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Term
| How do you find the tangent vector to a smooth curve at a point? |
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Definition
| find the first derivative of the vector that corresponds to the curve |
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Term
| How do you find the tangent line to a smooth curve at a point? |
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Definition
| find the line through the point parallel to the tangent vector |
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Term
| How do you find the unit tangent vector to a smooth curve? |
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Definition
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Term
What is the derivative of the following vector function:
u(t) + v(t) |
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Definition
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Term
What is the derivative of the following vector function:
cu(t) |
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Definition
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Term
What is the derivative of the following vector function:
f(t)*u(t) |
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Definition
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Term
What is the derivative of the following vector function:
u(t) • v(t) |
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Definition
u'(t) • v(t) + u(t) • v'(t)
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Term
What is the derivative of the following vector function:
u(t) x v(t) |
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Definition
| u'(t) x v(t) + u(t) x v'(t) |
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Term
What is the derivative of the following vector function:
u(f(t)) |
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Definition
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Term
| How do you find the length of a space curve given by a vector function r(t)? |
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Definition
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Term
| What is the definition of curvature? |
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Definition
| the change in the unit tangent vector as it moves around the curve |
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Term
| What is the formula for curvature in term of r'(t) and T'(t)? |
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Definition
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Term
| What is the formula for curvature in terms of r'(t) and r"(t)? |
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Definition
| k = |r'(t) x r"(t)|/|r'(t)|3 |
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Term
| What is the formula for curvature of a plane curve with equation y = f(x) |
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Definition
| k = |f"(x)|/[1 + (f'(x))2]3/2 |
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Term
| What is the formula for the unit normal vector of a smooth space curve r(t)? |
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Definition
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Term
| What is the formula for the binormal vector of a smooth space curve r(t)? |
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Definition
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Term
| What is the normal plane of a curve at a point? |
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Definition
| the plane whose normal vector is the unit tangent vector |
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Term
| What is the osculating plane of a curve at a point? |
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Definition
| the plane whose normal vector is the binormal vector |
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Term
| What is the osculating circle of a curve at a point? |
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Definition
| a circle that lies in the osculating plane and has a radius of 1/k |
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