Tangent Definition and 1000 Threads

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
The tangent line to a point on a differentiable curve can also be thought of as the graph of the affine function that best approximates the original function at the given point.Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
The word "tangent" comes from the Latin tangere, "to touch".

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  1. Rasalhague

    Tangent vectors as linear functionals on F(M)

    Let M be an n-dimensional manifold, with tangent spaces TpM for each point p in M. Let F(M) be the vector space of smooth functions M --> R, over R, with the usual definitions of addition and scaling. Tangent vectors in TM can be defined as linear functionals on F(M) (Fecko: Differential...
  2. L

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  3. R

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  4. T

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  5. J

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  6. Rasalhague

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  7. H

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  8. E

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  9. R

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  10. J

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  11. S

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  12. D

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  13. M

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  14. A

    Finding the Tangent Equation of a Scalar Field at (1,3,3) - Get Help Here

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  15. G

    Tangent of plane to a given surface

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  16. T

    Determining perpendicular tangent line

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  17. M

    Find the Proof: Tangent Lines & a Circle w/ Square Root of 3 Radius

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  18. A

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  19. S

    Help Solving Derivatives using Tables and Equation on Tangent Line

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  20. L

    Finding Normal & Tangent Vectors to Line: 3x-2y-4

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  21. D

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  22. A

    What determines the magnitude of a tangent vector?

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  23. C

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  24. D

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  25. H

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  26. H

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  27. Q

    Equation of Tangent Plane to Surface S at Point P(2,1,3)

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  28. T

    Does the System Define a Manifold and How to Find Tangent and Normal Spaces?

    Homework Statement OK I have a Differential Calculus exam next week and I do not understand about Differential Manifolds. We have been given some questions to practise, but I have no idea how to do them, past a certain point. For example 1. Study if the following system defines a manifold...
  29. L

    Horizontal Tangent Lines: Intersection of Cylinder and Plane

    Homework Statement Consider the plane z = x + 2y and the cylinder x^2 + y^2 = 1 (a) Find a vector function r(t) describing their intersection. (b) Find the points if any where the tangent to ~r is horizontal (c) Find an equation for the tangent line to ~r at each of these points.Homework...
  30. P

    Tangent line to curve, derivative.

    Homework Statement See the attachment. I feel like my answer is right, but I keep getting told I am wrong. This is the first time I have taken any math in 5 years so I am figuring I must not be simplifying enough? But I can't see what I need to simplify. Perhaps I have made a dumb error and...
  31. T

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  32. S

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    Greetings to all. :) This is my first time posting here, so if I do anything wrong, just tell me. :) On to the question. Alright, this question came in 3 parts. I've done the first 2 parts but have no clue on doing the third part. There are two equations (below) and a constant k...
  33. K

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  34. M

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  35. C

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  36. S

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  37. U

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  38. C

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    Homework Statement Find the slope of the tangent line to the curve: 2(x^2 + y^2)^2 = 25(x^2 - y^2) at the point (-3, -1) Homework Equations Implicit differentiation The Attempt at a Solution 2(x^2 + y^2)^2 = 25(x^2 - y^2) 1. 4(x^2 + y^2)(2x + 2y(dy/dx)) = 25(2x -...
  39. C

    Find the slope of the tangent line

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  40. U

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    Homework Statement Consider the curve x^2+xy+y^2=27 Homework Equations Find all points on the curve where the lines tangent to the curve are vertical The Attempt at a Solution I found dy/dy = (-2x-y)/x+2y) and I think I found the equations of lines visually to be x=6 and x=-6...
  41. K

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  42. M

    Does a Compact Manifold Imply a Compact Tangent Bundle?

    hello friends my question is: if we have M a compact manifold, do we have there necessarily TM compact ? thnx .
  43. W

    Line tangent to a curve (Vector style)

    Homework Statement Let L be the line tangent to the curve \vec{G}(t)=10\text{Cos}(t)\mathbf{i} + 10\text{Sin}(t)\mathbf{j} + 16t \mathbf{k} at the point (\frac{10}{\sqrt{2}}, \frac {10}{\sqrt{2}}, 4 \pi) Find the point at which L intersects the x-y plane.Homework Equations The Attempt at a...
  44. L

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    Homework Statement Consider the function f(x,y) = 4-x^2+3y^2 + y. Let S be the surface described by the equation z= f(x,y) where f(x,y) is given above. Find an equation for the plane tangent to S at the point (-1,0,3)The Attempt at a Solution Ok, SO i solved for the gradient of F...
  45. K

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  46. M

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  47. O

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  48. L

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  49. L

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  50. B

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