Welcome to MATH 2110Q Multivariable Calculus 1 Our goal is to - - PDF document

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welcome to math 2110q multivariable calculus
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Welcome to MATH 2110Q Multivariable Calculus 1 Our goal is to - - PDF document

Apr 04, 2023 263 likes •588 views

Welcome to MATH 2110Q Multivariable Calculus 1 Our goal is to extend calculus tools to account for multiple dimensions. For example, the NavierStokes equations for an incompressible fluid with 3D in space + 1D in time are: You should be

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Welcome to MATH 2110Q Multivariable Calculus

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Our goal is to extend calculus tools to account for multiple dimensions. For example, the Navier­Stokes equations for an incompressible fluid with 3D in space + 1D in time are: You should be able to "read" these equations at the end of the course.

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Let us stop for a bit and talk about the syllabus and other details of the course.

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3D Coordinates

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We can look at x­y, y­z, and x­z planes in 3D.

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Recall the four quadrants in 2D: In 3D there are eight "octants". You only need to know the "first octant" by number:

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Other octants:

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Projections: this concept can be generalized in many ways far outside the scope of this course. We will discuss "point" and "vector" projections. Projection of P=(a,b,c) onto the x­y plane is (a,b,0):

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Projection of P onto x­z plane is (a,0,c) Projection of P onto y­z plane is (0,b,c)

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  • SURFACES. 2D lines become surfaces in 3D.

There is no restriction on z in 3D for the relationship y=­x, so ALL z­values are included:

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Another common example would be y=1. In 3D, the surface y=1 is a plane parallel to the x­z plane but intersecting the y­axis at y=1;

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A more complicated example is a CYLINDER.

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Distance between points.

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Note Q is the projection of P onto the x­y plane here. The distance is just the size

  • f the z­coordinate of P.
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One useful idea is the set of all points that are EQUIDISTANT from a central point.

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We can also envision things like spherical VOLUMES;

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  • VECTORS. These are defined by precisely two

characteristics: (1) length and (2) magnitude.

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A vector can be VISUALIZED with the tail at any desired point; it is still the same vector.

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Vector addition.

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Multiplication by a scalar.

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Here are the algebraic rules; things are simply done "component­wise". Addition/subtraction: Scalar multiplication:

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Magnitude of a vector.

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Practice #1 Which of the following points is closest to the

  • rigin?
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Practice #2 Find the distance from P=(5, ­7, 4) to the y­z plane.

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Practice #3. Find an equation for a sphere with center (0,­1,2) and radius 7.

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Practice #4. Find the center and radius of the sphere

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Practice #5.