BSc: Mathematical Analysis II.s23 - Revision history

I.konyukhov: /* Final assessment */

2022-06-28T08:15:10Z

Final assessment

I.konyukhov: /* Final assessment */

2022-06-28T08:14:44Z

Final assessment

← Older revision		Revision as of 08:14, 28 June 2022
Line 219:		Line 219:

	=== Final assessment ===		=== Final assessment ===
−	~~'''~~Section 1~~'''~~	+	==== Section 1 ====
	# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}e^{-n\left(x^2+2\sin x\right)}</math>, <math display="inline">\Delta_1=(0;1]</math>, <math display="inline">\Delta_2=[1;+\infty)</math>;		# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}e^{-n\left(x^2+2\sin x\right)}</math>, <math display="inline">\Delta_1=(0;1]</math>, <math display="inline">\Delta_2=[1;+\infty)</math>;
	# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}\frac{\sqrt{nx^3}}{x^2+n^2}</math>, <math display="inline">\Delta_1=(0;1)</math>, <math display="inline">\Delta_2=(1;+\infty)</math>		# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}\frac{\sqrt{nx^3}}{x^2+n^2}</math>, <math display="inline">\Delta_1=(0;1)</math>, <math display="inline">\Delta_2=(1;+\infty)</math>

−	~~'''~~Section 2~~'''~~	+	==== Section 2 ====
	# Find all points where the differential of a function <math display="inline">f(x;y)=(5x+7y-25)e^{-x^2-xy-y^2}</math> is equal to zero.		# Find all points where the differential of a function <math display="inline">f(x;y)=(5x+7y-25)e^{-x^2-xy-y^2}</math> is equal to zero.
	# Show that function <math display="inline">\varphi=f\left(\frac xy;x^2+y-z^2\right)</math> satisfies the equation <math display="inline">2xz\varphi_x+2yz\varphi_y+\left(2x^2+y\right)\varphi_z=0</math>.		# Show that function <math display="inline">\varphi=f\left(\frac xy;x^2+y-z^2\right)</math> satisfies the equation <math display="inline">2xz\varphi_x+2yz\varphi_y+\left(2x^2+y\right)\varphi_z=0</math>.
Line 229:		Line 229:
	# <math display="inline">u=\left(y^2-x^2\right)e^{1-x^2+y^2}</math> on a domain given by inequality <math display="inline">x^2+y^2\leq4</math>;		# <math display="inline">u=\left(y^2-x^2\right)e^{1-x^2+y^2}</math> on a domain given by inequality <math display="inline">x^2+y^2\leq4</math>;

−	~~'''~~Section 3~~'''~~	+	==== Section 3 ====
	# Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral.		# Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral.
	# Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left\|a^2\leq x^2+y^2\leq 4a^2;\,\|x\|-y\geq0\right.\right\}</math>.		# Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left\|a^2\leq x^2+y^2\leq 4a^2;\,\|x\|-y\geq0\right.\right\}</math>.
	# Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">G=\left\{(x;y;z)\left\|x-1<y<x;\,1-x<y<2-x;\,1-x^2+y^2<z<y^2-x^2+2x\right.\right\}</math>.		# Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">G=\left\{(x;y;z)\left\|x-1<y<x;\,1-x<y<2-x;\,1-x^2+y^2<z<y^2-x^2+2x\right.\right\}</math>.

−	~~'''~~Section 4~~'''~~	+	==== Section 4 ====
	# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.		# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.
	# Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>;		# Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>;

I.konyukhov: /* Open access resources */

2022-06-23T21:03:34Z

Open access resources

← Older revision		Revision as of 21:03, 23 June 2022
Line 117:		Line 117:

	=== Open access resources ===		=== Open access resources ===
−	* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985	+	* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985 [https://www.cds.caltech.edu/~marsden/volume/Calculus/ link]
	* Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson		* Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson

I.konyukhov: /* Resources, literature and reference materials */

2022-06-23T21:02:44Z

Resources, literature and reference materials

← Older revision		Revision as of 21:02, 23 June 2022
Line 119:		Line 119:
	* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985		* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985
	* Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson		* Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson
−
−	=== Software and tools used within the course ===
−	* No.

	== Activities and Teaching Methods ==		== Activities and Teaching Methods ==

I.konyukhov: /* Final assessment */

2022-06-23T17:43:11Z

Final assessment

← Older revision		Revision as of 17:43, 23 June 2022
Line 238:		Line 238:

	'''Section 4'''		'''Section 4'''
		+	# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.
	# Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>;		# Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>;

I.konyukhov: /* Section 4 */

2022-06-23T17:42:55Z

Section 4

← Older revision		Revision as of 17:42, 23 June 2022
Line 220:		Line 220:
	# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.		# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.
	# Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at <math display="inline">A</math> and finishes at <math display="inline">B</math>: <math display="inline">\displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy</math>, <math display="inline">A(-2;-1)</math>, <math display="inline">B(0;3)</math>;		# Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at <math display="inline">A</math> and finishes at <math display="inline">B</math>: <math display="inline">\displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy</math>, <math display="inline">A(-2;-1)</math>, <math display="inline">B(0;3)</math>;
−

	=== Final assessment ===		=== Final assessment ===

I.konyukhov: /* Formative Assessment and Course Activities */

2022-06-23T17:42:06Z

Formative Assessment and Course Activities

← Older revision		Revision as of 17:42, 23 June 2022
Line 214:		Line 214:
	# Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: <math display="inline">\iint\limits_Df(x;y)\,dx\,dy</math> where <math display="inline">D=\left\{(x;y)\left\|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}</math>.		# Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: <math display="inline">\iint\limits_Df(x;y)\,dx\,dy</math> where <math display="inline">D=\left\{(x;y)\left\|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}</math>.
	# Represent integral <math display="inline">I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz</math> as iterated integrals with all possible (i.e. 6) orders of integration; <math display="inline">D</math> is bounded by <math display="inline">x=0</math>, <math display="inline">x=a</math>, <math display="inline">y=0</math>, <math display="inline">y=\sqrt{ax}</math>, <math display="inline">z=0</math>, <math display="inline">z=x+y</math>.		# Represent integral <math display="inline">I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz</math> as iterated integrals with all possible (i.e. 6) orders of integration; <math display="inline">D</math> is bounded by <math display="inline">x=0</math>, <math display="inline">x=a</math>, <math display="inline">y=0</math>, <math display="inline">y=\sqrt{ax}</math>, <math display="inline">z=0</math>, <math display="inline">z=x+y</math>.
−
⚫	==== Section 4 ====
	# Change order of integration in the iterated integral <math display="inline">\int\limits_0^{\sqrt2}dy\int\limits_y^{\sqrt{4-y^2}}f(x;y)\,dx</math>.		# Change order of integration in the iterated integral <math display="inline">\int\limits_0^{\sqrt2}dy\int\limits_y^{\sqrt{4-y^2}}f(x;y)\,dx</math>.
	# Find the volume of a solid given by <math display="inline">0\leq z\leq x^2</math>, <math display="inline">x+y\leq 5</math>, <math display="inline">x-2y\geq2</math>, <math display="inline">y\geq0</math>.		# Find the volume of a solid given by <math display="inline">0\leq z\leq x^2</math>, <math display="inline">x+y\leq 5</math>, <math display="inline">x-2y\geq2</math>, <math display="inline">y\geq0</math>.
		+
−	# Change into polar coordinates and rewrite the integral as a single integral: <math display="inline">\displaystyle\iint\limits_Gf\left(\sqrt{x^2+y^2}\right)\,dx\,dy</math>, <math display="inline">G=\left\{(x;y)\left\|x^2+y^2\leq x;\, x^2+y^2\leq y\right.\right\}</math>.
		⚫	==== Section 4 ====
		+	# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.
	# Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at <math display="inline">A</math> and finishes at <math display="inline">B</math>: <math display="inline">\displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy</math>, <math display="inline">A(-2;-1)</math>, <math display="inline">B(0;3)</math>;		# Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at <math display="inline">A</math> and finishes at <math display="inline">B</math>: <math display="inline">\displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy</math>, <math display="inline">A(-2;-1)</math>, <math display="inline">B(0;3)</math>;
		+

	=== Final assessment ===		=== Final assessment ===

I.konyukhov: /* Section 3 */

2022-06-23T17:40:59Z

Section 3

← Older revision		Revision as of 17:40, 23 June 2022
Line 214:		Line 214:
	# Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: <math display="inline">\iint\limits_Df(x;y)\,dx\,dy</math> where <math display="inline">D=\left\{(x;y)\left\|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}</math>.		# Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: <math display="inline">\iint\limits_Df(x;y)\,dx\,dy</math> where <math display="inline">D=\left\{(x;y)\left\|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}</math>.
	# Represent integral <math display="inline">I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz</math> as iterated integrals with all possible (i.e. 6) orders of integration; <math display="inline">D</math> is bounded by <math display="inline">x=0</math>, <math display="inline">x=a</math>, <math display="inline">y=0</math>, <math display="inline">y=\sqrt{ax}</math>, <math display="inline">z=0</math>, <math display="inline">z=x+y</math>.		# Represent integral <math display="inline">I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz</math> as iterated integrals with all possible (i.e. 6) orders of integration; <math display="inline">D</math> is bounded by <math display="inline">x=0</math>, <math display="inline">x=a</math>, <math display="inline">y=0</math>, <math display="inline">y=\sqrt{ax}</math>, <math display="inline">z=0</math>, <math display="inline">z=x+y</math>.
−	# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.

	==== Section 4 ====		==== Section 4 ====

I.konyukhov: /* Final assessment */

2022-06-23T17:40:36Z

Final assessment

← Older revision		Revision as of 17:40, 23 June 2022
Line 234:		Line 234:

	'''Section 3'''		'''Section 3'''
−	#
−
⚫	'''Section 4'''
	# Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral.		# Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral.
	# Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left\|a^2\leq x^2+y^2\leq 4a^2;\,\|x\|-y\geq0\right.\right\}</math>.		# Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left\|a^2\leq x^2+y^2\leq 4a^2;\,\|x\|-y\geq0\right.\right\}</math>.
	# Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">G=\left\{(x;y;z)\left\|x-1<y<x;\,1-x<y<2-x;\,1-x^2+y^2<z<y^2-x^2+2x\right.\right\}</math>.		# Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">G=\left\{(x;y;z)\left\|x-1<y<x;\,1-x<y<2-x;\,1-x^2+y^2<z<y^2-x^2+2x\right.\right\}</math>.
		+
		⚫	'''Section 4'''
	# Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>;		# Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>;

I.konyukhov: /* Final assessment */

2022-06-23T17:37:57Z

Final assessment

← Older revision		Revision as of 17:37, 23 June 2022
Line 237:		Line 237:

	'''Section 4'''		'''Section 4'''
		+	# Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral.
−	# Change order of integration in the iterated integral <math display="inline">\int\limits_0^{\sqrt2}dy\int\limits_y^{\sqrt{4-y^2}}f(x;y)\,dx</math>.
−	# ~~Find~~ the ~~volume of a solid given by~~ <math display="inline">0\~~leq z~~\~~leq~~ x^2</math>, <math display="inline">x+y\leq ~~5</math>~~, ~~<math display="inline">~~x-~~2y\geq2</math>, <math display="inline">~~y\geq0</math>.	+	# Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left\|a^2\leq x^2+y^2\leq 4a^2;\,\|x\|-y\geq0\right.\right\}</math>.
−	# ~~Change into polar coordinates and rewrite~~ the integral as a ~~single~~ ~~integral~~: <math display="inline">\displaystyle\~~iint~~\~~limits_Gf~~\left(~~\sqrt{~~x^2+y^2}\right)\,dx\,dy</math>, <math display="inline">G=\left\{(x;y)\left\|x~~^2+~~y^2\~~leq~~ x;\, x^2+y^2~~\leq~~ y\right.\right\}</math>.	+	# Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">G=\left\{(x;y;z)\left\|x-1<y<x;\,1-x<y<2-x;\,1-x^2+y^2<z<y^2-x^2+2x\right.\right\}</math>.
−	# ~~Having~~ ~~ascertained~~ ~~that~~ ~~integrand~~ ~~is an exact differential, calculate~~ the ~~integral along a piecewise~~ ~~smooth plain curve that starts at~~ <math display="inline">A</math> ~~and finishes at~~ <math display="inline">B</math>: <math display="inline">\~~displaystyle\int\limits_{\Gamma}\left(x^4~~+~~4xy^3~~\~~right)\,dx~~ +\~~left(6x^2y^2-5y^4\right)~~\,dy</math>, <math display="inline">~~A(-2;-1)~~</math>, <math display="inline">~~B(0;3)~~</math>;	+	# Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>;

	=== The retake exam ===		=== The retake exam ===

← Older revision		Revision as of 08:15, 28 June 2022
Line 222:		Line 222:
	# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}e^{-n\left(x^2+2\sin x\right)}</math>, <math display="inline">\Delta_1=(0;1]</math>, <math display="inline">\Delta_2=[1;+\infty)</math>;		# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}e^{-n\left(x^2+2\sin x\right)}</math>, <math display="inline">\Delta_1=(0;1]</math>, <math display="inline">\Delta_2=[1;+\infty)</math>;
	# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}\frac{\sqrt{nx^3}}{x^2+n^2}</math>, <math display="inline">\Delta_1=(0;1)</math>, <math display="inline">\Delta_2=(1;+\infty)</math>		# Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}\frac{\sqrt{nx^3}}{x^2+n^2}</math>, <math display="inline">\Delta_1=(0;1)</math>, <math display="inline">\Delta_2=(1;+\infty)</math>
−
	==== Section 2 ====		==== Section 2 ====
	# Find all points where the differential of a function <math display="inline">f(x;y)=(5x+7y-25)e^{-x^2-xy-y^2}</math> is equal to zero.		# Find all points where the differential of a function <math display="inline">f(x;y)=(5x+7y-25)e^{-x^2-xy-y^2}</math> is equal to zero.
Line 228:		Line 227:
	# Find maxima and minima of function <math display="inline">u=2x^2+12xy+y^2</math> under condition that <math display="inline">x^2+4y^2=25</math>. Find the maximum and minimum value of a function		# Find maxima and minima of function <math display="inline">u=2x^2+12xy+y^2</math> under condition that <math display="inline">x^2+4y^2=25</math>. Find the maximum and minimum value of a function
	# <math display="inline">u=\left(y^2-x^2\right)e^{1-x^2+y^2}</math> on a domain given by inequality <math display="inline">x^2+y^2\leq4</math>;		# <math display="inline">u=\left(y^2-x^2\right)e^{1-x^2+y^2}</math> on a domain given by inequality <math display="inline">x^2+y^2\leq4</math>;
−
	==== Section 3 ====		==== Section 3 ====
	# Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral.		# Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral.
	# Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left\|a^2\leq x^2+y^2\leq 4a^2;\,\|x\|-y\geq0\right.\right\}</math>.		# Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left\|a^2\leq x^2+y^2\leq 4a^2;\,\|x\|-y\geq0\right.\right\}</math>.
	# Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">G=\left\{(x;y;z)\left\|x-1<y<x;\,1-x<y<2-x;\,1-x^2+y^2<z<y^2-x^2+2x\right.\right\}</math>.		# Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">G=\left\{(x;y;z)\left\|x-1<y<x;\,1-x<y<2-x;\,1-x^2+y^2<z<y^2-x^2+2x\right.\right\}</math>.
−
	==== Section 4 ====		==== Section 4 ====
	# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.		# Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>.