2nd year mathematics dr. rafea dakhil 2017-2018 lectures

2nd

Year Mathematics Dr. Rafea Dakhil

2017-2018 Lectures

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1. Areas in Polar Coordinates

This section shows how to calculate areas of plane regions, lengths of curves, and areas of surfaces

of revolution in polar coordinates.

1.1. Areas in the Plane

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1.2. Plane Area between Two Curves

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1.3. Length of a Polar Curve

(2)

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1.4. Area of a Surface of Revolution

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Vectors

1. Vector in Plane

A quantity such as force, displacement, or velocity is called a vector and is represented by a

directed line segment.

𝐴 = 𝑂𝐴⃑⃑⃑⃑ ⃑⃑ = 𝑎𝑖 + 𝑏𝑗 (𝑉𝑒𝑐𝑡𝑜𝑟)

|𝐴| = √𝑎2 + 𝑏2 (𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟)

𝛼 + 𝛽 =𝜋

2

cos 𝛼 = sin 𝛽

𝑖, 𝑗 are the fundamental unit vectors

𝑈𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 = �⃑⃑� =𝐴

|𝐴|=

𝑎𝑖 + 𝑏𝑗

|𝐴|

∴ �⃑⃑� = (cos 𝛼) 𝑖 + (cos 𝛽)𝑗

𝑤ℎ𝑒𝑟𝑒, cos 𝛼 =𝑎

|𝐴| ; cos 𝛽 =

𝑏

|𝐴|

Vector Algebra Operations

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Notes

1. 𝑉𝑒𝑐𝑡𝑜𝑟 (𝑎𝑖 + 𝑏𝑗) 𝑖𝑠 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 (𝑏𝑖 − 𝑎𝑗).

2. 𝑉𝑒𝑐𝑡𝑜𝑟 (𝑎𝑖 + 𝑏𝑗) 𝑖𝑠 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑡ℎ𝑎𝑡 ℎ𝑎𝑠 𝑎𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (𝑎𝑥 + 𝑏𝑦 = 𝑐).

3. 𝑉𝑒𝑐𝑡𝑜𝑟 (𝑏𝑖 − 𝑎𝑗) 𝑖𝑠 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑡ℎ𝑎𝑡 ℎ𝑎𝑠 𝑎𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (𝑎𝑥 + 𝑏𝑦 = 𝑐).

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Note:

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The Dot Product

Dot products are also called inner or scalar products because the product results in a scalar, not a

vector.

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Writing a Vector as a Sum of Orthogonal Vectors

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The Cross Product

The vector is orthogonal to both u and v because it is a scalar n.

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Triple Scalar or Box Product

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Equations of Lines and Planes in Space

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The Distance from a Point to a Line in Space

There are two methods to find the distance from a point to a line in the space as follows:

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PARTIAL DERIVATIVES

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The Chain Rule

Chain Rule for Functions of Two Independent Variables

Chain Rule for Functions of Three Independent Variables

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Directional Derivatives and Gradient Vectors

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H.W:

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Tangent Planes and Normal Lines

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Multiple Integrals

1. Double Integrals y

x

R

dy

dx

dA

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Finding Limits of Integration

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Moments and Centres of Mass for Thin Flat Plates

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Double Integrals in Polar Form

Finding Limits of Integration

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2nd year mathematics dr. rafea dakhil 2017-2018 lectures

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