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Page 1 Vector Basics ? Position vector :- The position vector of the point P(x, y, z) in the space is ?? = ?? ?? + ?? ?? + ?? ?? ?? = ?? 2 + ?? 2 + ?? 2 ? In parametric form, ?? = ?? ?? ?? + ?? ?? ?? + ?? (?? ) ?? ? Let, ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? , ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? ?? .?? = ?? ?? cos ?? . ?? = ?? 1 ?? 1 + ?? 2 ?? 2 + ?? 3 ?? 3 ?? × ?? = ?? |?? | sin( ?? . ?? ) ?? , where n is vector of unit length perpendicular to the plane containing ?? & ?? . ? ?? × ?? = ?? ?? ?? ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 Page 2 Vector Basics ? Position vector :- The position vector of the point P(x, y, z) in the space is ?? = ?? ?? + ?? ?? + ?? ?? ?? = ?? 2 + ?? 2 + ?? 2 ? In parametric form, ?? = ?? ?? ?? + ?? ?? ?? + ?? (?? ) ?? ? Let, ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? , ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? ?? .?? = ?? ?? cos ?? . ?? = ?? 1 ?? 1 + ?? 2 ?? 2 + ?? 3 ?? 3 ?? × ?? = ?? |?? | sin( ?? . ?? ) ?? , where n is vector of unit length perpendicular to the plane containing ?? & ?? . ? ?? × ?? = ?? ?? ?? ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 Vector Basics ? Area of ??????? = 1 2 ???? × ???? = 1 2 ?? × ?? ? Area of ??????? = 1 2 ?? ?? × ???? = 1 2 (?? - ?? ) × ( ?? - ?? ) ? Area of parallelogram = | ?? × ?? | ? Scalar triple product :- ?? × ?? . ?? = ?? . ?? × ?? = [ ?? ?? ?? ] = ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ? Vector triple product :- ?? × ?? × ?? = ?? . ?? ?? - ?? . ?? ?? Page 3 Vector Basics ? Position vector :- The position vector of the point P(x, y, z) in the space is ?? = ?? ?? + ?? ?? + ?? ?? ?? = ?? 2 + ?? 2 + ?? 2 ? In parametric form, ?? = ?? ?? ?? + ?? ?? ?? + ?? (?? ) ?? ? Let, ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? , ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? ?? .?? = ?? ?? cos ?? . ?? = ?? 1 ?? 1 + ?? 2 ?? 2 + ?? 3 ?? 3 ?? × ?? = ?? |?? | sin( ?? . ?? ) ?? , where n is vector of unit length perpendicular to the plane containing ?? & ?? . ? ?? × ?? = ?? ?? ?? ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 Vector Basics ? Area of ??????? = 1 2 ???? × ???? = 1 2 ?? × ?? ? Area of ??????? = 1 2 ?? ?? × ???? = 1 2 (?? - ?? ) × ( ?? - ?? ) ? Area of parallelogram = | ?? × ?? | ? Scalar triple product :- ?? × ?? . ?? = ?? . ?? × ?? = [ ?? ?? ?? ] = ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ? Vector triple product :- ?? × ?? × ?? = ?? . ?? ?? - ?? . ?? ?? Vector Differentiation ? Let ?? ?? = ?? (?? ) then, ?? ?? ???? = lim ??? ?0 ?? ?? +??? - ?? (?? ) ??? ? If t is a time variable then ?? ?? ???? represents a velocity vector. 1. ?? ?? ???? is a vector in direction of tangent to the curve at that point. 2. If ?? (?? ) is constant in magnitude then ?? . ?? ?? ???? = 0 3. If ?? (?? ) has constant direction then, ?? × ?? ?? ???? = 0 Page 4 Vector Basics ? Position vector :- The position vector of the point P(x, y, z) in the space is ?? = ?? ?? + ?? ?? + ?? ?? ?? = ?? 2 + ?? 2 + ?? 2 ? In parametric form, ?? = ?? ?? ?? + ?? ?? ?? + ?? (?? ) ?? ? Let, ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? , ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? ?? .?? = ?? ?? cos ?? . ?? = ?? 1 ?? 1 + ?? 2 ?? 2 + ?? 3 ?? 3 ?? × ?? = ?? |?? | sin( ?? . ?? ) ?? , where n is vector of unit length perpendicular to the plane containing ?? & ?? . ? ?? × ?? = ?? ?? ?? ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 Vector Basics ? Area of ??????? = 1 2 ???? × ???? = 1 2 ?? × ?? ? Area of ??????? = 1 2 ?? ?? × ???? = 1 2 (?? - ?? ) × ( ?? - ?? ) ? Area of parallelogram = | ?? × ?? | ? Scalar triple product :- ?? × ?? . ?? = ?? . ?? × ?? = [ ?? ?? ?? ] = ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ? Vector triple product :- ?? × ?? × ?? = ?? . ?? ?? - ?? . ?? ?? Vector Differentiation ? Let ?? ?? = ?? (?? ) then, ?? ?? ???? = lim ??? ?0 ?? ?? +??? - ?? (?? ) ??? ? If t is a time variable then ?? ?? ???? represents a velocity vector. 1. ?? ?? ???? is a vector in direction of tangent to the curve at that point. 2. If ?? (?? ) is constant in magnitude then ?? . ?? ?? ???? = 0 3. If ?? (?? ) has constant direction then, ?? × ?? ?? ???? = 0 Vector Differentiation ? Vector differential operator :- ?? (nebla) ?? = ?? ?? ???? + ?? ?? ?? ?? + ?? ?? ?? ?? ? Gradient of a scalar function :- Let ?? (?? , ?? , ?? ) be a differentiable scalar point function then gradient of scalar is denoted by grad ?? or ?? ?? = ?? ?? ?? ???? + ?? ?? ?? ???? + ?? ?? ?? ???? ? Where, ?? ?? is vector normal to surface ?? . ? Unit vector normal to surface ?? can be given as ?? ?? |?? ?? | . Page 5 Vector Basics ? Position vector :- The position vector of the point P(x, y, z) in the space is ?? = ?? ?? + ?? ?? + ?? ?? ?? = ?? 2 + ?? 2 + ?? 2 ? In parametric form, ?? = ?? ?? ?? + ?? ?? ?? + ?? (?? ) ?? ? Let, ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? , ?? = ?? 1 ?? + ?? 2 ?? + ?? 3 ?? ?? .?? = ?? ?? cos ?? . ?? = ?? 1 ?? 1 + ?? 2 ?? 2 + ?? 3 ?? 3 ?? × ?? = ?? |?? | sin( ?? . ?? ) ?? , where n is vector of unit length perpendicular to the plane containing ?? & ?? . ? ?? × ?? = ?? ?? ?? ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 Vector Basics ? Area of ??????? = 1 2 ???? × ???? = 1 2 ?? × ?? ? Area of ??????? = 1 2 ?? ?? × ???? = 1 2 (?? - ?? ) × ( ?? - ?? ) ? Area of parallelogram = | ?? × ?? | ? Scalar triple product :- ?? × ?? . ?? = ?? . ?? × ?? = [ ?? ?? ?? ] = ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ?? 1 ?? 2 ?? 3 ? Vector triple product :- ?? × ?? × ?? = ?? . ?? ?? - ?? . ?? ?? Vector Differentiation ? Let ?? ?? = ?? (?? ) then, ?? ?? ???? = lim ??? ?0 ?? ?? +??? - ?? (?? ) ??? ? If t is a time variable then ?? ?? ???? represents a velocity vector. 1. ?? ?? ???? is a vector in direction of tangent to the curve at that point. 2. If ?? (?? ) is constant in magnitude then ?? . ?? ?? ???? = 0 3. If ?? (?? ) has constant direction then, ?? × ?? ?? ???? = 0 Vector Differentiation ? Vector differential operator :- ?? (nebla) ?? = ?? ?? ???? + ?? ?? ?? ?? + ?? ?? ?? ?? ? Gradient of a scalar function :- Let ?? (?? , ?? , ?? ) be a differentiable scalar point function then gradient of scalar is denoted by grad ?? or ?? ?? = ?? ?? ?? ???? + ?? ?? ?? ???? + ?? ?? ?? ???? ? Where, ?? ?? is vector normal to surface ?? . ? Unit vector normal to surface ?? can be given as ?? ?? |?? ?? | . Vector Differentiation ? Directional derivative :- The directional derivative of differentiable scalar function ?? (?? , ?? , ?? ) in the direction of ?? is given by, ?? ?? . ?? |?? | ? Let ?? = ?? , then, ? D.D. = ?? ?? . ?? | ?? | = ( ?? ?? ?? ???? + ?? ?? ?? ???? + ?? ?? ?? ???? ). ?? = ?? ?? ???? ? Angle between surfaces :- It is the angle between the normal to the surfaces at the point of intersection. Let ?? be the angle between the surfaces ?? 1 ?? , ?? , ?? = ?? 1 & ?? 2 ?? , ?? , ?? = ?? 2 then, cos ?? = ?? Ø 1 ?? ?? 2 |?? Ø 1 | |?? ?? 2 |Read More
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FAQs on PPT: Vector Calculus - Engineering Mathematics - Civil Engineering (CE)
| 1. What is vector calculus? |  |
Ans. Vector calculus is a branch of mathematics that deals with vector fields and their derivatives. It involves the study of vectors and vector-valued functions, along with operations such as differentiation and integration of vectors.
| 2. How is vector calculus used in physics? | |
Ans. Vector calculus is extensively used in physics to describe and analyze various physical phenomena. It is used to determine quantities like velocity, acceleration, and force, which are typically represented as vectors. Vector calculus is also used in electromagnetism, fluid mechanics, and other areas of physics.
| 3. What are the basic operations in vector calculus? | |
Ans. The basic operations in vector calculus include taking the gradient, divergence, and curl of a vector field. The gradient represents the rate of change of a scalar function, while the divergence measures the amount of a vector field's source or sink at a given point. The curl measures the rotation or circulation of a vector field.
| 4. How can vector calculus be applied to real-life problems? | |
Ans. Vector calculus can be applied to real-life problems in various fields such as engineering, physics, and computer graphics. For example, it can be used to analyze fluid flow in pipes, optimize the design of structures, model electromagnetic fields, and simulate realistic motion in computer animations.
| 5. Are there any practical applications of vector calculus in everyday life? | |
Ans. While vector calculus may not have direct practical applications in everyday life for most people, its principles and concepts have indirect influences. For instance, vector calculus plays a role in the development of GPS technology, computer graphics used in movies and video games, and even weather forecasting models.
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